Theta and eta polynomials in geometry, Lie theory, and combinatorics
Harry Tamvakis







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Theta and eta polynomials]
Theta and eta polynomials in geometry, Lie theory, and combinatorics



 
University of Maryland, Department of
Mathematics, William E. Kirwan Hall, 4176 Campus Drive, 
College Park, MD 20742, USA
harryt@math.umd.edu

[2010]Primary 14M15; Secondary 05E05, 14N15





The classical Schur polynomials form a natural basis for the ring of
symmetric polynomials and have geometric significance since they
represent the Schubert classes in the cohomology ring of
Grassmannians.  Moreover, these polynomials enjoy rich combinatorial
properties. In the last decade, an exact analogue of this picture has
emerged in the symplectic and orthogonal Lie types, with the Schur
polynomials replaced by the theta and eta polynomials of Buch, Kresch,
and the author. This expository paper gives an overview of what is
known to date about this correspondence, with examples.




[
    Harry Tamvakis
    April 14, 2020
==================






§ INTRODUCTION


The structure of the cohomology ring of the Grassmannian (m,n) of
m dimensional linear subspaces of ^n was first explored by
Schubert <cit.>, Pieri <cit.>, and Giambelli <cit.>.  The ring
^*((m,n),) has an additive basis of Schubert classes, coming
from the cell decomposition of (m,n), and there is a natural
choice of multiplicative generators for this ring, namely, the special
Schubert classes. Giambelli showed that a general Schubert class,
which may be indexed by a partition, can be expressed as a
Jacobi-Trudi determinant <cit.> whose entries are special
classes. It soon became apparent (see for example <cit.>) that the
resulting algebra is closely connected to the theory of Schur
polynomials. The latter polynomials were originally defined by Cauchy
<cit.>, and studied by many others since then, motivated to a large
extent by their applications to the representation theory of the
symmetric and general linear groups <cit.>, and related
combinatorics.

The theory of theta and eta polynomials, by contrast, has its origins
in geometry, and specifically in the desire to extend the
aforementioned work of Giambelli to the cohomology of symplectic and
orthogonal Grassmannians. The first steps in this direction were taken
in the 1980s by Hiller and Boe <cit.> and Pragacz <cit.>. They
proved Pieri and Giambelli formulas for the Grassmannians of maximal
isotropic subspaces, with the Schubert classes indexed by strict
partitions and the Jacobi-Trudi determinants replaced by Schur
Pfaffians <cit.>.

In 2008, Buch, Kresch, and the author announced a series of works
<cit.> which went beyond these hermitian
symmetric (or cominuscule) examples. Two crucial insights from
op. cit. were the identification of the correct set of special
Schubert class generators to employ, and the realization of the
essential role that Young's raising operators <cit.> play in
the theory. The latter becomes clear only when one attempts to
understand the cohomology of nonmaximal isotropic
Grassmannians.  Our papers introduced k-strict partitions to
index the Schubert classes and theta polynomials to represent
them, in both the classical and quantum cohomology rings of general
symplectic and odd orthogonal Grassmannians. Three years later, the
companion papers <cit.> dealt with the even orthogonal case, using
typed k-strict partitions and eta polynomials.

Theta and eta polynomials can be viewed as symmetric polynomials for
the action of the corresponding Weyl group, but this is not obvious
from their definition, and was pointed out only recently <cit.>. In
fact, a substantial part of the theory of Schur polynomials can be
extended to the world of theta and eta polynomials, but this requires
a change in perspective, as well as the introduction of new techniques
of proof. It turns out that these objects can be applied to solve the
(equivariant) Giambelli problem for the classical Lie groups, that is,
to obtain intrinsic polynomial representatives for the (equivariant)
Schubert classes on any classical G/P-space – so, any (isotropic)
partial flag manifold <cit.>. The resulting formulas
(<ref>), (<ref>), and (<ref>) are stated
using solely the language of Lie theory.

The goal of this expository paper is to illustrate the correspondence
between Schur polynomials and theta/eta polynomials in the case of
single polynomials, where the story is most complete. There is ample
room for further interesting connections to be found, and the reader
is likely to discover more by just asking for the theta/eta
analogue of their favorite statement about Schur polynomials.  We have
stopped short of discussing extensions of some of these results to the
theory of degeneracy loci and equivariant cohomology, quantum
cohomology, and K-theory. 

This article is organized as follows. The Schur, theta, and eta
polynomials are featured in Sections <ref>, <ref>,
and <ref>, respectively. Each of these is split into parallel
subsections on initial definitions and Pieri rules, the cohomology of
Grassmannians, symmetric polynomials, algebraic combinatorics, and the
cohomology of partial flag manifolds. Finally, Section <ref>
contains historical notes and references.

It is a pleasure to thank the Hellenic Mathematical Society for the
invitation to lecture in the First Congress of Greek Mathematicians in
Athens, June 2018. The present paper is a more detailed and expanded
version of my talk there.




§ SCHUR POLYNOMIALS





 §.§ Definition using raising operators


We will define the classical Schur polynomials by using the
Jacobi-Trudi formula, but rewritten in the language of Young's raising
operators. Let u_1, u_2,… be a sequence of commuting
independent variables, and set u_0:=1 and u_i:=0 for
i<0. Throughout the paper, :=(_1,_2,…) will denote
an integer sequence with only finitely many nonzero terms
_i. For any such , we let u_:=u__1u__2⋯,
which is a monomial in the variables u_i.


Given any integer sequence α and i<j, we define

    R_ij(α) := (α_1,…,α_i+1,…,α_j-1,
    …).
 
A raising operator R is any monomial in the R_ij's. For 
any such R, we let R u_ := u_R(), where, as is customary, 
we regard the operator R as acting on the index , and not on 
the monomial u_. 

An integer sequence  is a partition if _i≥_i+1≥ 0 for each i≥ 1. If  is a partition, we let
|| denote the sum of all its parts _i, and the length
ℓ() be the number of i such that _i≠ 0.  A partition
 may be represented using its Young diagram of boxes, placed in
left justified rows, with _i boxes in the i-th row, for each
i≥ 1. The diagram of the conjugate partition  is
obtained by taking the transpose of the diagram of .  The
containment relation ⊂μ between two partitions is defined
using their respective diagrams; in this case the set-theoretic
difference μ is the skew diagram μ/. For example,
we illustrate below the diagram of the partition (4,3,3,2) and the
skew diagram (4,3,3,2)/(3,1,1).

    < g r a p h i c s >



The Schur polynomial s_(u) is 
defined by the raising operator formula

    s_(u) := ∏_i<j (1-R_ij)  u_.

We regard the products in formulas such as (<ref>) as 
being over all pairs (i,j) with j≤ N for some N greater than 
or equal to the length ℓ(). The result will always be independent
of any such N, so we may assume that N=ℓ().



For any partition =(a,b) with two parts a and b, we have

    s_a,b(u)=(1-R_12)   u_a,b = 
    u_a,b - u_a+1,b-1 = u_au_b-u_a+1u_b-1 = 
    |[   u_a u_a+1; u_b-1   u_b ]|.



The formula of Example <ref> generalizes: for any partition 
of length ℓ, we have

    s_(u) =  (u__i+j-i)_1≤ i,j≤ℓ.

The equivalence of (<ref>) with (<ref>) is a formal
consequence of the Vandermonde identity

    ∏_1≤ i<j≤ℓ(_i-_j)= (_i^ℓ-j)_1≤ i,j≤ℓ


The monomials u_ and the polynomials s_(u) as  runs
over all partitions form two -bases of the graded polynomial ring
:=[u_1,u_2,…]. In addition to the Giambelli formula
(<ref>), these two bases of  interact via the Pieri rule. To state the latter, recall that a skew diagram is a
horizontal strip (resp. a vertical strip) if it does not
contain two boxes in the same column (resp. row).  For any partition
 and p≥ 0, we then have the Pieri rule

    u_p· s_(u) = ∑_μ s_μ(u)

with the sum over all partitions μ⊃ such that μ/
is a horizontal strip with p boxes.

 We have

    u_3· s_2,2,1(u) = s_5,2,1(u)+s_4,2,2(u)+s_4,2,1,1(u)+
    s_3,2,2,1(u).





 §.§ Cohomology of Grassmannians
 

Let (m,n) be the Grassmannian of all m-dimensional complex
linear subspaces of ^n. The general linear group _n() acts
transitively on (m,n), and the stabilizer of the point ⟨
e_1,…, e_m⟩ under this action is the parabolic subgroup
P_m of matrices in _n() in the block form

    (
    [ * *; 0 * ])

where the lower left block is an (n-m)× m zero matrix. 
It follows that 

    (m,n)=_n/P_m

and hence that (m,n) is a complex manifold of dimension m(n-m). 
Furthermore, (m,n) has the structure of an algebraic variety,
and the same is true of all the geometric objects which appear in this paper.

The Grassmannian (m,n) has a natural decomposition into Schubert
cells ^∘_, one for each partition  whose diagram is
contained in an m× (n-m) rectangle. The Schubert variety
_ is the closure of the cell ^∘_, and is an algebraic
subvariety of (m,n) of complex codimension equal to ||.
Concretely, if e_1,…,e_n is the canonical basis of 
^n, and F_i is the -linear span of e_1,…,e_i, then

    _ := {V∈(m,n) | (V∩ F_n-m+i-_i)≥ i,  1≤ i ≤ m}.

If [_] denotes the cohomology class of _, then the cell
decomposition of (m,n) implies that there is an isomorphism of
abelian groups

    ^*((m,n),) ≅⊕_ [_].


For every integer p with 1≤ p ≤ n-m, the variety _p is
the locus of subspaces V in (m,n) which meet the subspace
F_n-m+1-p nontrivially.  The varieties _p are the special Schubert varieties, and their cohomology classes [_p]
are the special Schubert classes.  Let Q→(m,n) denote the
universal quotient vector bundle over (m,n). Then for each
integer p≥ 0, the p-th Chern class c_p(Q) of Q is equal to
[_p] in ^2p((m,n),).  We can now state the Giambelli formula

    [_] = s_(c(Q))

where  is the conjugate partition of , and the Chern
class polynomial s_(c(Q)) is obtained from
s_(u) by performing the substitutions u_p↦ c_p(Q)
for each integer p.  The discussion in <ref> and
equation (<ref>) imply that the Pieri rule

    [_p]· [_] = ∑_μ [_μ]

also holds in ^*((m,n),), where the sum is over all indexing
partitions μ containing  such that μ/ is a horizontal
strip with p boxes.



 §.§ Symmetric polynomials



Fix an integer n≥ 1 and let X_n:=(x_1,…,x_n), where the 
x_i are independent variables. The Weyl group S_n of _n 
acts on the polynomial ring [X_n] by permuting the variables, 
and the invariant subring is the ring Λ_n:=[X_n]^S_n
of symmetric polynomials. Two important families of elements of 
Λ_n are the elementary symmetric polynomials e_p(X_n) and 
the complete symmetric polynomials h_p(X_n). These are defined 
the by the generating function equations 

    ∑_p=0^∞ e_p(X_n)t^p = ∏_i=1^n(1+x_it)   and  ∑_p=0^∞ h_p(X_n)t^p = ∏_i=1^n(1-x_it)^-1,

respectively, where t is a formal variable. The fundamental theorem of 
symmetric polynomials states that 

    Λ_n=[e_1(X_n),…,e_n(X_n)].


For each partition , the Schur polynomial s_(X_n) is obtained
from s_(u) by making the substitution u_p↦ h_p(X_n) for every
integer p. In this way, we obtain the Jacobi-Trudi formula

    s_(X_n)=∏_i<j (1-R_ij)  h_(X_n) = 
    (h__i+j-i(X_n))_1≤ i,j≤ℓ(),

where we have set h_:=h__1h__2⋯ 
for every integer sequence . If we similarly let 
e_:=e__1e__2⋯, then we have the dual Jacobi-Trudi formula

    s_(X_n)=∏_i<j (1-R_ij)  e_(X_n) = 
    (e__i+j-i(X_n))_1≤ i,j≤ℓ().

We deduce that 

    Λ_n=⊕_ s_(X_n)

where the sum is over all partitions =(_1,…,_n) of 
length at most n.

The classical definition of Schur polynomials is as a quotient of
alternant determinants. For each integer k≥ 1, let
δ_k:=(k,…,1,0), and, for any integer vector
=(_1,…,_n), let x^:=x_1^_1⋯
x_n^_n.  Define the alternating operator  on [X_n] by

    (f):=∑_∈ S_n (-1)^ℓ()(f),

where ℓ() denotes the length of the permutation . 
Then we have

    s_(X_n)=. (x_i^_i+n-j)_i,j(x_i^n-j)_i,j = 
    . (x^+δ_n-1)(x^δ_n-1).


We restate equation (<ref>) in a form closer to its analogue
for theta polynomials. For each r≥ 1, embed S_r into S_r+1
by adjoining the fixed point r+1. Let S_∞=∪_rS_r denote
the corresponding infinite symmetric group of bijections :_>0→_>0 such that _i=i for all but finitely many i. Here,
and in the sequel, _i denotes the value (i), for each i
≥ 1.  The code of  is the the sequence
γ=γ() with γ_i:=#{j>i | _j<_i}. The
     shape of  is the partition =() whose parts
     are the nonzero entries γ_i arranged in weakly decreasing
     order.


An n-Grassmannian permutation  is an element of S_∞ such
that _i < _i+1 for each i≠ n. The shape of any such
 is the partition :=(_n-n,…,_1-1). Conversely, any
partition =(_1,…,_n) of length at most n corresponds
to a unique n-Grassmannian permutation  with ()=.


Let  be an n-Grassmannian element of S_∞ with
corresponding partition (). If _0:=(n,n-1,…,1)
denotes the longest permutation in S_n, then observe that
(_0)=δ_n-1 and (_0)=()+δ_n-1.
Therefore, we have

    s_()(X_n)=. (x^(_0))(x^(_0)).





 §.§ Algebraic combinatorics



Although not strictly necessary, in this subsection we will extend our
family of variables to be an infinite sequence X:=(x_1,x_2,…), 
and work with formal power series instead of polynomials. Define
h_p(X) using the generating function expansion

    ∑_p=0^∞h_p(X)t^p = ∏_i=1^∞ (1-x_it)^-1,

and, for any partition , the Schur function

    s_(X):=∏_i<j(1-R_ij)  h_(X).


A tableau T on the shape 
is a filling of the boxes of  with positive integers, so that
the entries are weakly increasing along each row from left to right
and strictly increasing down each column. The content vector 
c(T)=(n_1,n_2,…) of T has n_j equal to the number of 
entries j in T. We then have the tableau formula

    s_(X)= ∑_Tx^c(T)

with the sum over all tableaux T of shape . Equation (<ref>)
shows that s_(X) is a formal power series with nonnegative integer
coefficients, which have a combinatorial interpretation.


A Young tableaux of shape =(2,1) is of the form

    < g r a p h i c s >

where the positive integers a,b,c satisfy a≤ b and a<c. We
therefore have

    s_2,1(X)= ∑_a≤ ba<cx_ax_bx_c = 
    ∑_a≠ bx_a^2x_b + 2∑_a<b<cx_ax_bx_c.



In order to represent the Schubert classes not only on Grassmannians, but 
on any partial flag variety, we require a generalization of (<ref>) 
which involves certain symmetric functions defined by Stanley. 
The group S_∞ is generated by the simple transpositions 
s_i=(i,i+1) for i≥ 1. A reduced word of a permutation 
in S_∞ is a sequence a_1⋯ a_ℓ of positive integers
such that =s_a_1⋯ s_a_ℓ and ℓ is minimal, so (by
definition) equal to the length ℓ() of . 

The nilCoxeter algebra _n of the symmetric group S_n is
the free associative algebra with unit generated by the elements
ξ_1,…,ξ_n-1, modulo the relations

    [         ξ_i^2             =             0        i≥ 1 ;;        ξ_iξ_j             =        ξ_jξ_i    |i-j|≥ 2 ;;   ξ_iξ_i+1ξ_i             = ξ_i+1ξ_iξ_i+1         i≥ 1. ]

For any ∈ S_n, choose a reduced word a_1⋯ a_ℓ for
 and define ξ_ := ξ_a_1…ξ_a_ℓ. Then the ξ_
for ∈ S_n are well defined, independent of the choice of reduced 
word a_1⋯ a_ℓ, and form a free -basis of
_n. We denote the coefficient of ξ_∈_n in the
expansion of the element ζ∈_n by ⟨ζ,⟩.
We therefore have 

    ζ = ∑_∈ S_n⟨ζ,⟩ ξ_,

for all ζ∈_n.

Recall that t denotes an indeterminate and define

    A(t) := (1+t ξ_n-1)(1+t ξ_n-2)⋯ 
    (1+t ξ_1)
      ; 
    
    A(X) := A(x_1)A(x_2)⋯

and a function G_(X) for ∈ S_n by

    G_(X) := ⟨ A(X), ⟩.

It turns out that G_(X) is symmetric in the x_i variables; it 
is called a Stanley symmetric function. Clearly, G_
has nonnegative integer coefficients. 

When  is the n-Grassmannian permutation associated to a
partition  of length at most n, then G_(X)=s_(X). More
generally, for any permutation , when G_ is expanded in the
basis of Schur functions, we have

    G_(X) = ∑_  :  || = ℓ()c^_ s_(X)

for some nonnegative integers c^_.  That is, the symmetric
function G_ is Schur positive. There exist several
different combinatorial interpretations of the coefficients
c^_, but the most important one for our purposes uses the transition trees of Lascoux and Schützenberger, as subsequently defined.

We say that a permutation =(_1,_2,…) has a descent at position i≥ 1 if _i>_i+1. For i<j, let
t_ij denote the transposition which interchanges i and j.  For
any permutation ∈ S_∞, the transition tree T()
of  is constructed as follows.  The tree T() is a rooted
tree with nodes given by permutations of the same length ℓ(),
and root .  If =1 or  is Grassmannian, then set
T():={}. Otherwise, let r be the largest descent of ,
and set 

    s := max(j>r | _j < _r).
 
Define

    I():={i  | 1≤ i < r   and  ℓ(
    t_rst_ir) = ℓ() }

and let

    Ψ():={ t_rst_ir | i∈ I()}   if I()≠∅, 
    Ψ(1×)    otherwise.

We define T() recursively, by
joining  by an edge to each v∈Ψ(), and attaching to
each v∈Ψ() its tree T(v). One can show that T() is a
finite tree whose leaves are all Grassmannian permutations. Moreover,
the Stanley coefficient c^_ in equation (<ref>) is equal
to the number of leaves of shape  in the transition tree T()
associated to .



The transition tree for the permutation =(2,1,5,4,3) is shown below.

    < g r a p h i c s >

It follows that

    G_21543(X)=s_3,1(X)+s_2.2(X)+s_2,1,1(X).






 §.§ Cohomology of flag manifolds


Let {e_1,…,e_n} denote the standard basis of
E:=^n and let F_i=⟨ e_1,…, e_i⟩ 
be the subspace spanned by the first i vectors of this basis.
The group G=_n() acts transitively on the space of all complete 
flags in E, and the stabilizer of the flag F_ is the Borel
subgroup B of invertible upper triangular matrices. If T⊂ B
denotes the maximal torus of diagonal matrices, then the Weyl group 
W=S_n can be identified with N_G(T)/T.

The parabolic subgroups P of _n with P⊃ B
correspond to sequences a_1<⋯ < a_p of positive
integers with a_p<n. For any such P, the manifold _n/P
parametrizes partial flags of subspaces

    0 = E_0 ⊂ E_1 ⊂⋯⊂ E_p ⊂ E=^n

with (E_r) = a_r for each r∈ [1,p]. We agree that
E_r and E will also denote the corresponding
tautological vector bundles over _n/P. The associated
parabolic subgroup W_P of S_n is generated by the
simple transpositions s_i for i∉{a_1,…, a_p}. 

There is a canonical presentation of the cohomology ring of _n/B,
which gives geometric significance to the variables which appear in
Section <ref>. Let _n denote the ideal of [X_n]
generated by the homogeneous elements of positive degree in
Λ_n, so that _n=⟨
e_1(X_n),…,e_n(X_n)⟩. We then have ring isomorphism

    ^*(_n/B) ≅[X_n]/_n

which maps each variable x_i to -c_1(E_i/E_i-1). Moreover, for any 
parabolic subgroup P of _n, the projection map _n/B→_n/P induces an injection ^*(_n/P)↪^*(_n/B) of cohomology rings, and we have

    ^*(_n/P) ≅[X_n]^W_P/_n^P,

where [X_n]^W_P denotes the W_P-invariant subring of [X_n], 
and _n^P is the ideal of [X_n]^W_P generated by 
e_1(X_n),…,e_n(X_n).

Consider the set

    W^P := {∈ S_n | ℓ( s_i) = ℓ()+1, ∀  i ∉{a_1,…, a_p},  i<n}

of minimal length W_P-coset representatives in S_n. We have
a decomposition

    _n = ⋃_∈ W^PB P

and for each ∈ W^P, the B-orbit of P in _n/P is
the Schubert cell ^∘_:=B P/P. Let _ be the
closure of ^∘_ in _n/P, and set
_:=__0.  The Schubert class [_] is the
cohomology class of _ in ^2ℓ()(_n/P,). We
thus obtain an isomorphism of abelian groups

    ^*(_n/P,) ≅⊕_∈ W^P[_]

which generalizes (<ref>). 

If V_1 and V_2 are complex vector bundles over a manifold M, define
cohomology classes ϕ_p by the generating function equation

    ∑_p=0^∞ϕ_pt^p = c_t(V_2^*)/c_t(V_1^*),

where c_t(V_i^*)=1-c_1(V_i)t+c_2(V_i)t^2-⋯ is the Chern
polynomial of V_i^*, for i=1,2. Given any partition , the
polynomial s_(V_1-V_2) is obtained from s_(u) via the
substitution u_p↦ϕ_p for every p∈, so that

    s_(V_1-V_2):= ∏_i<j (1-R_ij) ϕ_.


Recall that E_r for r∈ [1,p] and E denote the tautological and
trivial rank n vector bundles over _n/P, respectively. For
any ∈ W^P, we then have

    [_] = ∑_ c^_ 
    s_^1(E-E_1) s_^2(E_1-E_2)⋯ s_^p(E_p-1-E_p)

in ^*(_n/P,), where the sum is over all sequences of
partitions =(^1,…,^p) and the 
coefficients c^_ are given by 

    c^_ := ∑_u_1⋯ u_p = 
    c_^1^u_1⋯ c_^p^u_p

summed over all factorizations u_1⋯ u_p= such that
ℓ(u_1)+⋯ + ℓ(u_p)=ℓ() and u_j(i)=i for all j>1
and i ≤ a_j-1. The nonnegative integers c^u_i_^i which
appear in the summands in (<ref>) agree with the Stanley
coefficients from equation (<ref>).  When p=1, the partial flag
manifold _n/P is the Grassmannian (a_1,n), and formula
(<ref>) specializes to equation (<ref>).




Let P=B be the Borel subgroup, so that the flag manifold _n/B
parametrizes complete flags of subspaces 0=E_0⊂ E_1 ⊂⋯⊂ E_n =^n. For each i∈ [1,n], let
x_i:=-c_1(E_i/E_i-1).  Then for any partition , we have

    s_(E_i-1-E_i) = 
    
    x_i^r    if =r≥ 0, 
    
    0    otherwise.

Let A_i(t):=(1+t ξ_n-1)(1+t ξ_n-2)⋯ (1+t ξ_i) in
_n[t], employing the notation of Section <ref>. Define the Schubert polynomial _(X_n) by

    _(X_n):= ⟨ A_1(x_1)⋯ A_n-1(x_n-1), ⟩.

It is then straightforward to show that formula (<ref>) is 
equivalent to the statement that, for any permutation ∈ S_n, 
we have [_]=_(X_n) in ^*(_n/B,).








§ THETA POLYNOMIALS
 




 §.§ Definition and Pieri rule



Fix a nonnegative integer k. A partition  is called k-strict if no part greater than k is repeated, that is,
_i>k⇒_i >_i+1. A strict partition is the same
as a 0-strict partition. For a general k-strict partition ,
we define the operator

    R^ := ∏_i<j (1-R_ij)∏__i+_j > 2k+j-i
    (1+R_ij)^-1

where the first product is
over all pairs i<j and second product is over pairs i<j such that
_i+_j > 2k+j-i. The theta polynomial ^(k)_(u) of level 
k is defined by 

    ^(k)_(u) := R^ u_.

We will write _(u) for ^(k)_(u) when the level k is
understood.




(a) Suppose that =(a,b) has two parts a and b with a+b>2k+1.
Then we have

    _a,b(u)    = 1-R_12/1+R_12  u_a,b = 
    (1-2R_12+2R_12^2-2R_12^3+⋯)   u_a,b
       = u_au_b-2u_a+1u_b-1 +2u_a+2u_b-2-2u_a+3u_b-3+⋯.


(b) If _i≤ k for each i, then 

    Θ_(u) = ∏_i<j(1-R_ij)  u_  = 
     (u__i+j-i)_1≤ i,j≤ℓ().


(c) If _i>k for all nonzero parts _i, then

    Θ_(u) = ∏_i<j1-R_ij/1+R_ij u_ = 
    (__i,_j(u))_1≤ i<j≤ℓ'

where ℓ' is the least positive even integer such that 
ℓ'≥ℓ().


Example <ref> shows that as  varies, the theta polynomial
_ interpolates between the determinant (<ref>) and
the Pfaffian (<ref>). The equality of Example <ref>(c) is
a formal consequence of Schur's Pfaffian identity

    ∏_1≤ i<j ≤ℓ'_i-_j/_i+_j =
    (_i-_j/_i+_j)_1≤ i,j ≤ℓ'.



Let k=2 and =(5,2,1). Then we have

    ^(2)_5,2,1(u)     = 
    1-R_12/1+R_12(1-R_13)(1-R_23)   u_5,2,1
       = (1-2R_12+2R_12^2 - 2 R_12^3)(1-R_13-R_23) 
      u_5,2,1
       = u_5u_2u_1 -u_5u_3 -2 u_6u_1^2 + u_6u_2+2u_7u_1.





Recall that =[u_1,u_2,…], and let ^(k) be the quotient
of  by the ideal of relations

    1-R_12/1+R_12  u_p,p = 
    u_p^2 + 2∑_i=1^p(-1)^i u_p+iu_p-i= 0
       for  p > k.

Then the monomials u_ and the polynomials _(u) as  runs
over all k-strict partitions form two -bases of the graded ring
^(k). To state the Pieri rule for the _(u), which holds 
modulo the relations (<ref>), we need some further definitions. 

We say that the box in row r and column c of a k-strict
partition λ is k-related to the box in row r' and
column c' if |c-k-1|+r = |c'-k-1|+r'.  For example, the two grey
boxes in the following partition are k-related.

    0.65shiftedrel


For any two k-strict partitions λ and μ, we have a
relation λμ if |μ|=||+p and μ is
obtained by removing a vertical strip from the first k columns of
λ and adding a horizontal strip to the resulting diagram, so
that

(1) if one of the first k columns of μ has the same number of
boxes as the same column of λ, then the bottom box of this
column is k-related to at most one box of μ∖λ; and

(2) if a column of μ has fewer boxes than the same column of
λ, then the removed boxes and the bottom box of μ in this
column must each be k-related to exactly one box of μ∖λ, and these boxes of μ∖λ must all lie in the same row.

If λμ, we let  be the set of boxes of
μ in columns k+1 and higher which are not
mentioned in (1) or (2). Define N(λ,μ) to be the number
of connected components of  which do not have a box in column
k+1. Here we consider that two boxes are connected if they have at 
least one vertex in common.

For any k-strict partition  and p≥ 0, we then
have the Pieri rule

    u_p·_(u) = ∑_μ 2^N(,μ)_μ(u)

in ^(k), where the sum over all k-strict partitions μ such that
μ.

 

(a) When k=1, we have the following equality in ^(1):

    u_3·_2,1(u) = 2 _6(u) + 4 _5,1(u) + _4,2(u)+
    2 _4,1,1(u) +_3,2,1(u).

 
(b) If ||+p ≤ k, then 

    u_p·_(u) = ∑_μ_μ(u)

holds in ^(k), where the sum is over all partitions μ⊃ such that |μ|=||+p and μ/ is a horizontal strip.

(c) If _i>2k for all nonzero parts _i and p≥ 0 is arbitrary, 
then

    u_p·_(u) = ∑_μ 2^N(,μ)_μ(u)

holds in ^(k), where the sum is over all strict partitions μ⊃ such that |μ|=||+p and μ/ is a horizontal strip,
and N(,μ) equals the number of connected components of
μ/ which do not meet the first column.


In harmony with Example <ref>, Example <ref> illustrates
that as p and  vary, the rule for the product u_p·_ interpolates between the Pieri rule (<ref>) for
Schur polynomials and (<ref>), which is the Pieri rule for
the Schur Q-functions (see Example <ref>).




 §.§ Cohomology of Grassmannians
 



Equip the vector space ^2n with the nondegenerate skew-symmetric
bilinear form ( , ) defined by the conditions (e_i,e_j)=0 for
i+j≠ 2n+1 and (e_i,e_2n+1-i)=1 for 1≤ i ≤ n. The
symplectic group _2n() is the subgroup of _2n()
consisting of those elements g such that (gv_1,gv_2)=(v_1,v_2),
for every v_1,v_2∈^2n. We say that a linear subspace V of
^2n is isotropic if the restriction of ( , ) to V
vanishes identically. Since the form is nondegenerate, we have
(V)≤ n for any isotropic subspace V. If V is isotropic
and (V)=n then we call V a Lagrangian subspace.

Fix an integer k with 0≤ k ≤ n-1. The isotropic Grassmannian
=(n-k,2n) parametrizes all isotropic linear subspaces of
dimension n-k in ^2n. The group _2n acts
transitively on (n-k,2n), and the stabilizer of a fixed isotropic
(n-k)-plane under this action is a maximal parabolic subgroup P_k
of _2n, so that (n-k,2n) = _2n/P_k.

The Schubert cells ^∘_ in (n-k,2n) are indexed by
the k-strict partitions whose diagrams are contained in an
(n-k)× (n+k) rectangle. The Schubert variety _ is the
closure of the ^∘_, and has codimension || in .
If F_i denotes the -linear span of e_1,…,e_i for each
i∈ [1,2n], then

    _λ := {V ∈|(V ∩
       F_p_j(λ)) ≥ j    ∀  1 ≤ j ≤ n-k },

where the strictly increasing index function {p_j(λ)}_1≤
  j ≤ n-k is defined by

    p_j(λ) := n+k+j-λ_j - #{i<j  | _i+_j>2k+j-i }.

If [_] denotes the cohomology class of _ in
^2||(,), then we have a group isomorphism

    ^*((n-k,2n),) ≅⊕_ [_].


The varieties _p for 1≤ p ≤ n+k are the special
  Schubert varieties.
Let Q→(n-k,2n) denote the universal quotient vector bundle
over , which has rank n+k. For every integer p≥ 0, the
p-th Chern class c_p(Q) is equal to the special Schubert
  class [_p] in ^2p((n-k,2n),). We now have the Giambelli formula

    [_] = _(c(Q))

where the Chern class polynomial _(c(Q)) is obtained from
_(u) by performing the substitutions u_p↦ c_p(Q)
for each integer p. Moreover, the Pieri rule 

    [_p]· [_] = ∑_μ 2^N(,μ)[_μ]

holds in ^*((n-k,2n),), where the sum is over all partitions 
μ such that μ and the diagram of μ fits 
in an (n-k)× (n+k) rectangle.


The ring ^(k) is naturally isomorphic to the stable cohomology
ring

    (_k) = lim_⟵^*((n-k,2n),)

of the isotropic Grassmannian . This is the inverse limit in
the category of graded rings of the directed system

    ⋯←^*((n-k,2n),) ←^*((n+1-k,2n+2),) ←⋯

Under this isomorphism, the variables u_p map to the Chern classes
c_p(Q) of the universal quotient bundle Q over . If S
denotes the tautological subbundle of the trivial rank 2n vector
bundle over (n-k,2n), then the symplectic form ( , ) gives a
pairing S⊗ Q→𝒪, which in turn produces an
injection S ↪ Q^*. Using the Whitney sum formula
c(S)c(Q)=1, we therefore obtain

    c(Q^*)c(Q) = c(Q^*)c(S)^-1=c(Q^*/S).

The relations (<ref>) arise from this, 
using the fact that c_r(Q^*/S)=0 for r>2k.





 §.§ Symmetric polynomials



Let c:=(c_1,c_2,…) be a sequence of commuting independent
variables, set c_0:=1 and c_p=0 for p<0, and for every integer
sequence , let c_:=c__1c__2⋯. For any raising
operator R, we let R  c_:=c_R. Consider the graded ring
Γ which is the quotient of the polynomial ring [c] modulo
the ideal generated by the relations

    c_p^2+2∑_i=1^p(-1)^ic_p+ic_p-i=0,    for all p≥ 1.

The ring Γ is isomorphic to ^(0) and to the stable cohomology ring

    () = lim_⟵^*((n,2n),)

of the Lagrangian Grassmannian , with the variables
c_p mapping to the Chern classes
c_p(Q) of the universal quotient bundle Q→.

The Weyl group of the symplectic group _2n is the hyperoctahedral group of signed permutations on the set
{1,…,n}, which is the semidirect product S_n ⋉_2^n
of S_n with _2^n={± 1}^n. We use a bar over an entry to
denote a negative sign; thus w=(2,3,1) maps (1,2,3) to
(-2,-3,1). The group W_n is generated by the simple transpositions
s_i=(i,i+1) for 1≤ i ≤ n-1 and the sign change s_0 which
satisfies s_0(1)=1 and s_0(j)=j for all j≥ 2. A reduced word of an element w∈ W_n is a sequence a_1⋯
a_ℓ of nonnegative integers of minimal length ℓ such that
w=s_a_1⋯ s_a_ℓ. The number ℓ is called the length of w, and denoted by ℓ(w).

Fix n≥ 1 and let X_n:=(x_1,…,x_n), as in Section <ref>.
There is a natural action of W_n on Γ[X_n] which extends the 
action of S_n on [X_n], defined as follows. 
The simple transpositions s_i for i∈ [1,n-1] 
act by interchanging x_i and x_i+1 while leaving all the remaining
variables fixed. The reflection s_0 satisfies s_0(x_1)=-x_1 and 
s_0(x_j)=x_j for all j ≥ 2, while

    s_0(c_p) := c_p+2∑_j=1^p x_1^jc_p-j  for all p≥ 1.

If t denotes a formal variable which is fixed by s_0, then we 
express equation (<ref>) using generating functions as

    s_0(∑_p=0^∞ c_pt^p) = 1+x_1t/1-x_1t(∑_p=0^∞ c_pt^p).


For every integer p, define an element ^nc_p of Γ[X_n] by

    ^nc_p:=∑_j=0^p c_p-je_j(X_n),    for p≥ 1

and let Γ^(n) := [^nc_1, ^nc_2, …]. Let
Γ[X_n]^W_n denote the subring of W_n-invariants in
Γ[X_n]. We claim that the generators ^nc_p of
Γ^(n) lie in Γ[X_n]^W_n. Indeed, we clearly have
s_j(^nc_p)=^nc_p for each j≥ 1, while

    s_0(∑_p=0^∞^nc_pt^p)    =
    s_0(∑_p=0^∞ c_pt^p ·∏_j=1^n(1+x_jt))
    
       = 1+x_1t/1-x_1t(∑_p=0^∞ c_pt^p
    )· (1-x_1t)∏_j=2^n(1+x_jt) = ∑_p=0^∞^nc_pt^p.

In fact, there is an equality

    Γ[X_n]^W_n=Γ^(n)= [^nc_1, ^nc_2, …].


The map which sends u_p to ^nc_p for every integer p induces a ring 
isomorphism ^(n)≅Γ^(n). We therefore have

    Γ^(n)=⊕_ _(X_n)

where the sum is over all n-strict partitions , and the
polynomial _(X_n) is obtained from the theta polynomial
^(n)_(u) by making the substitution u_p↦^nc_p for
all p. In other words, we have

    _(X_n)=^(n)_(X_n) := R^  (^nc)_,

where we set (^nc)_:=^nc__1^nc__2⋯ for any
integer sequence , and the raising operators R_ij in R^
act on the subscripts  as usual. 


For each r≥ 1, we embed W_r in W_r+1 by adding the element
r+1 which is fixed by W_r, and set W_∞ :=∪_r W_r.  Let
w∈ W_∞ be a signed permutation.  To simpify the notation, 
let w_i denote the value w(i), for each i≥ 1. Define a strict
partition μ(w) whose parts are the absolute values of the negative
entries of w, arranged in decreasing order. Let the A-code of
w be the sequence γ with γ_i:=#{j>i | w_j<w_i},
and define a partition δ(w) whose parts are the nonzero
entries γ_i arranged in weakly decreasing order. The shape of w is the partition (w):=μ(w)+ν(w), where
ν(w):=δ(w) is the conjugate of δ(w). One can
show that the length ℓ(w) of w is equal to |(w)|.


(a) An n-Grassmannian signed permutation w is an element of W_∞ such
that w_1>0 and w_i < w_i+1 for each i≠ n. The shape of any
such w is the n-strict partition  satisfying

    _i= 
    n+|w_n+i|    if w_n+i<0, 
    #{r≤ n  :  w_r> w_n+i}   if w_n+i>0.

Conversely, any n-strict partition  corresponds to a unique
n-Grassmannian permutation w with (w)=. 

(b) Let w_0:=(1,…,n) be
the longest element of W_n. Then μ(w_0)=δ_n, ν(w_0)=
δ_n-1, and (w_0)=δ_n+δ_n-1=(2n-1,2n-3,…,
1).


Let w be an n-Grassmannian element of W_∞ with
corresponding partition λ(w), and let w_0 be the longest
element of W_n.  For any integer sequence  and composition
β, let ^ c_:=
^_1c__1^_2c__2⋯, and set R_ij^
c_:= ^ c_R_ij for each i<j. Consider the multi-Schur Pfaffian

    ^ν(ww_0)Q_(ww_0) := ∏_i<j1-R_ij/1+R_ij ^ν(ww_0)c_(ww_0).

Define the alternating operator ' on Γ[X_n] by

    '(f):=∑_w∈ W_n (-1)^ℓ(w)w(f),

where ℓ(w) is the length of the signed permutation w. 
We then have

    _(w)(X_n) 
    =(-1)^n(n+1)/2. 
    '(^ν(ww_0)Q_(ww_0))'(x^(w_0))

in Γ[X_n].




 §.§ Algebraic combinatorics



The combinatorial formulas discussed in this section require another
incarnation of the ring Γ, using the formal power series known
as Schur Q-functions. Fix a nonnegative integer k, set
X_k:=(x_1,…,x_k) and let Z:=(z_1,z_2,…) be a sequence of
variables. For any integer p, define _p=_p(Z ;X_k) by the
generating function equation

    ∑_p=0^∞_pt^p = ∏_i=1^∞1+z_it/1-z_it∏_j=1^k(1+x_jt).

By definition, for every k-strict partition ,
the theta polynomial _(Z ;X_k) is obtained from
^(k)_(u) by making the substitution u_p↦_p
for every integer p. In other words, we have

    _(Z ;X_k):= R^ _

where, for any integer sequence ,
_=__1__2⋯ and the raising operators
R_ij in R^ are applied to _ as usual.  Note that
_(Z ;X_k) is a formal power series in the Z variables and a
polynomial in the variables x_1,…,x_k.



Suppose that k=0 and  is a strict partition. Then the formal
power series Q_(Z):=_(Z) is a Schur Q-function. The
map which sends c_p to Q_p:=Q_p(Z) for every integer p gives an
isomorphism between the ring Γ defined in Section <ref>
and the ring [Q_1,Q_2,…] of Schur Q-functions.



We wish to describe a tableau formula for _(Z ;X_k) which is
analogous to the expression (<ref>) for the Schur
functions. The following key observation is used to define the
relevant tableaux. In the Pieri rule (<ref>) for the product
u_p·_(u), all partitions μ which appear on the right
hand side may be written as μ=(p+r,ν) for some integer r≥ 0
and k-strict partition ν with ν⊂. Moreover, if p
is sufficiently large (for example p>||+2k) and we write

    u_p·_(u) = ∑_r,ν2^n(/ν) _(p+r,ν)(u),

with the sum over integers r≥ 0 and k-strict partitions
ν⊂ with |ν| = ||-r, then the ν which appear in
(<ref>) and the exponents n(/ν) are independent of
p. If this is the case, so that (p+r,ν) for
any p>||+2k, then we say that /ν is a k-horizontal
  strip.

Let  and μ be any two k-strict partitions with μ⊂.
A k-tableau T of shape /μ is a sequence of k-strict
partitions

    μ = ^0⊂^1⊂⋯⊂^r=

such that ^i/^i-1 is a k-horizontal strip for 1≤ i≤
r.  We represent T by a filling of the boxes in /μ with
positive integers which is weakly increasing along each row and down
each column, such that for each i, the boxes in T with entry i
form the skew diagram ^i/^i-1. For any
k-tableau T we define n(T):=∑_i n(^i/^i-1) and let c(T)
denote the content vector of T.

Let P denote the ordered alphabet
{1'<2'<⋯<k'<1<2<⋯}.  We say that the symbols
1',…,k' are marked, while the rest are unmarked.  A
k-bitableau U of shape  is a filling of the boxes in
the diagram of  with elements of P which is weakly
increasing along each row and down each column, such that the marked
entries are strictly increasing along each row, and the unmarked
entries form a k-tableau T. We define

    n(U):=n(T)   and  
    (zx)^c(U):= z^c(T) ∏_j=1^k x_j^m_j

where m_j denotes the number of times that j' appears in U.  For
any k-strict partition , we then have the tableau formula

    _(Z ;X_k) = ∑_U 2^n(U)(zx)^c(U)

where the sum is over all k-bitableaux U of shape . Using the
tableau formula (<ref>) for Schur polynomials, we can rewrite
equation (<ref>) as 

    _(Z ;X_k) = ∑_μ⊂∑_T 2^n(T)z^c(T) s_μ(X_k)

with the sums over all partitions μ⊂ and k-tableaux T
of shape /μ, respectively.




If k=0 and  is a strict partition, then (<ref>) 
becomes

    Q_(Z) = ∑_T 2^n(T)z^c(T)

summed over all 0-tableaux T of shape . Equation (<ref>)
is a tableau formula for the Schur Q-functions.



(a) For any integer p≥ 0, we have 

    _p(Z ;X_k) = ∑_j=0^p ∑_||=p-j 2^#z^ s_1^j(X_k)
    =∑_j=0^p Q_p-j(Z)e_j(X_k)

where the second sum is over all compositions  with ||=p-j, 
and # denotes the number of indices i such that _i≠ 0.

(b) Assume that k≥ 1. Then we have
 
    _1^p(Z ;X_k) = ∑_j=0^p∑_||=p-j 2^#z^ s_j(X_k)
    =∑_j=0^p Q_p-j(Z)h_j(X_k).





The group W_∞ is generated by the reflections s_i for 
i≥ 0, and these generators are used to define reduced words and the 
length of signed permutations as in Section <ref>.
The nilCoxeter algebra _n of the hyperoctahedral group W_n 
is the free associative algebra with unit
generated by the elements ξ_0,…,ξ_n-1, modulo the relations

    [         ξ_i^2             =             0        i≥ 0 ;;        ξ_iξ_j             =        ξ_jξ_i    |i-j|≥ 2 ;;   ξ_iξ_i+1ξ_i             = ξ_i+1ξ_iξ_i+1        i≥ 1 ;;  ξ_0ξ_1ξ_0ξ_1             = ξ_1ξ_0ξ_1ξ_0. ]

For any w∈ W_n, choose a reduced word a_1⋯ a_ℓ for
w and set ξ_w := ξ_a_1…ξ_a_ℓ. Then the ξ_w
are well defined and form a free -basis of
_n. We denote the coefficient of ξ_w∈_n in the
expansion of the element ζ∈_n by ⟨ζ,w⟩.

Let t be an independent variable, define

    C(t) := (1+t ξ_n-1)⋯ (1+t ξ_1)(1+tξ_0)(1+tξ_0)(1+tξ_1) 
    ⋯ (1+tξ_n-1)

and let C(Z):=C(z_1)C(z_2)⋯. Choose an integer k with 0≤
k<n and set A(X_k):=A(x_1)⋯ A(x_k). For any w∈ W_n, the
(restricted) type C mixed Stanley function J_w(Z ;X_k) is
defined by

    J_w(Z ;X_k) := ⟨ C(Z)A(X_k), w⟩.

Clearly, J_w is a power series in the Z variables and a polynomial
in the X_k variables, and has nonnegative integer coefficients. One
can show that J_w is symmetric in the Z and X_k variables
separately.

When w is the k-Grassmannian permutation associated to a
k-strict partition , then 

    J_w(Z ;X_k)=_(Z ;X_k).

This equality may be generalized as follows.  We say that a signed
permutation w=(w_1,…,w_n) has a descent at position
i≥ 1 if w_i>w_i+1, and a descent at i=0 if and only if
w_1<0. A signed permutation w∈ W_n is increasing up to k
if it has no descents less than k. This condition is automatically
satisfied if k=0, and for positive k it means that 0 < w_1 < w_2
< ⋯ < w_k. For any element w∈ W_n which is increasing up to
k, we have

    J_w(Z ;X_k) = ∑_e^w_ _(Z ;X_k)

summed over k-strict partitions  with ||=ℓ(w). The
integers e^w_ are nonnegative, that is, the function
J_w(Z ;X_k) is theta positive when w is increasing up to
k. These coefficients have only one known combinatorial
interpretation, which is given below.

For positive integers i<j, define the reflections t_ij
and t_ij,t_ii in W_∞ by their right actions

    (w_1,…,w_i,…,w_j,…) t_ij   = 
    (w_1,…,w_j,…,w_i,…), 
    
    (w_1,…,w_i,…,w_j,…) t_ij   = 
    (w_1,…,w_j,…,w_i,…),   and
    
    (w_1,…,w_i,…) t_ii   = 
    (w_1,…,w_i,…),
 
and let t_ji := t_ij.  For any w∈ W_∞ which
is increasing up to k, we construct a rooted tree T^k(w) with root
w and whose nodes are elements of W_∞ as follows. Let r be
the largest descent of w. If w=1 or r=k, then set
T^k(w):={w}. Otherwise, let s := max(i>r | w_i < w_r) and
Φ(w) := Φ_1(w)∪Φ_2(w), where

    Φ_1(w) := {wt_rst_ir | 1≤ i < r   and  ℓ(wt_rst_ir) = ℓ(w) }, 
    Φ_2(w) := 
    {wt_rst_ir | i≥ 1   and ℓ(wt_rst_ir) = ℓ(w) }.

We define T^k(w) recursively, by joining w by an edge to each
v∈Φ(w), and attaching to each v∈Φ(w) its tree
T^k(v). The finite tree T^k(w) is the k-transition tree of
w, and its leaves are all k-Grassmannian elements of W_∞.
The coefficient e^w_ in (<ref>) is equal to the number of
leaves of T^k(w) which have shape .


Suppose that k=1. The 1-transition tree for the signed permutation
w=(3,1,2,5,4) in W_5 is shown below.

    < g r a p h i c s >

It follows that

    J_31254(Z ;X_1)=_4(Z ;X_1)+2 _3,1(Z ;X_1)+_2,1,1(Z ;X_1).













 §.§ Cohomology of flag manifolds


Let {e_1,…,e_2n} denote the standard symplectic basis of
E:=^2n and let F_i=⟨ e_1,…, e_i⟩ be the
subspace spanned by the first i vectors of this basis, as in Section
<ref>.  The group G=_2n acts transitively on the space
of all complete isotropic flags in E, and the stabilizer of the flag
F_ is a Borel subgroup B of G. Let T⊂ B denote the
maximal torus of diagonal matrices in G, and the Weyl group
W=N_G(T)/T≅ W_n.

The parabolic subgroups P of _2n with P⊃ B
correspond to sequences a_1<⋯ < a_p of nonnegative 
integers with a_p<n. For any such P, the manifold :=_2n/P
parametrizes partial flags of subspaces

    E_ : 0 ⊂ E_p ⊂⋯⊂ E_1 ⊂ E=^2n

with E_1 isotropic and (E_r) = n-a_r for each r∈ [1,p]. 
As usual, E_r and E will also denote the corresponding
tautological vector bundles over . The associated
parabolic subgroup W_P of W_n is generated by the
simple reflections s_i for i∉{a_1,…, a_p}. 


There is a canonical presentation of the cohomology ring of _2n/B,
which gives geometric significance to the variables which appear in
Section <ref>. Let ^(n) denote the ideal of Γ[X_n]
generated by the homogeneous elements of positive degree in
Γ^(n), so that ^(n)=⟨^nc_1,^nc_2,…⟩. We then have ring isomorphism

    ^*(_2n/B,) ≅Γ[X_n]/^(n)

which maps each variable c_p to c_p(E/E_n) and x_i to
c_1(E_n+1-i/E_n-i) for 1≤ i ≤ n. Furthermore, for any
parabolic subgroup P of _2n, the projection map _2n/B→_2n/P induces an injection ^*(_2n/P,)↪^*(_2n/B,) on cohomology rings, and we have

    ^*(,) ≅Γ[X_n]^W_P/^(n)_P,

where Γ[X_n]^W_P denotes the W_P-invariant subring of
Γ[X_n], and _P^(n) is the ideal of Γ[X_n]^W_P 
generated by ^nc_1,^nc_2,…

We have a decomposition

    _2n = ⋃_w ∈ W^PBw P

where 

    W^P := {w∈ W_n | ℓ(ws_i) = ℓ(w)+1, ∀  i ∉{a_1,…, a_p},  i<n}

is the set of minimal length W_P-coset representatives in W_n.
For each w ∈ W^P, the B-orbit of wP in  is the Schubert cell ^∘_w:=Bw P/P. The Schubert variety
_w is the closure of ^∘_w in _2n/P. Then
_w:=_w_0w has codimension ℓ(w) in , and its
cohomology class [_w] is a Schubert class. The cell
decomposition of  implies that there is an isomorphism of abelian
groups

    ^*(,) ≅⊕_w∈ W^P[_w]

which generalizes (<ref>). 


Recall that E_r for r∈ [1,p] and E denote the tautological and
trivial vector bundles over , of rank n-a_r and 2n,
respectively. For any w∈ W^P, we then have

    [_w] = ∑_ f^w_ ^(a_1)_^1(E-E_1) s_^2(E_1-E_2)⋯ s_^p(E_p-1-E_p)

in ^*(,), where the sum is over all sequences of
partitions =(^1,…,^p) with ^1 being 
a_1-strict, and the 
coefficients f^w_ are given by 

    f^w_ := ∑_u_1⋯ u_p = w
    e_^1^u_1c_^2^u_2⋯ c_^p^u_p

summed over all factorizations u_1⋯ u_p= w such that
ℓ(u_1)+⋯ + ℓ(u_p)=ℓ(w), u_j∈ S_∞ for j≥
2, and u_j(i)=i for all j>1 and i ≤ a_j-1.  The
nonnegative integers e^u_1_^1 and c^u_i_^i which
appear in the summands in (<ref>) are the same as the ones in
equations (<ref>) and (<ref>), respectively.  When p=1, the
partial flag manifold  is the isotropic Grassmannian
(n-a_1,2n), and formula (<ref>) specializes to equation
(<ref>). In general, the polynomial ^(a_1)_^1(E-E_1) 
in (<ref>) is defined by pulling back the polynomial 
^(a_1)_^1(c(E/E_1)) under the natural projection map 
→(n-a_1,2n) which sends a partial flag E_ to E_1.




Let P=B be the Borel subgroup, so that the flag manifold
_2n/B parametrizes flags of subspaces 0⊂ E_n ⊂⋯⊂ E_1 ⊂ E=^2n with E_1 Lagrangian and
(E_i) = n+1-i for each i∈ [1,n]. For each i, let
x_i:=-c_1(E_i/E_i+1), and observe that since E/E_1≅ E_1^*,
for any integer p, we have c_p(E/E_1) = e_p(x_1,…,x_n) =
e_p(X_n), using the definition of Chern classes. For every strict
partition , define the Q-polynomial
Q_(X_n) by the formula

    Q_(X_n):=∏_i<j1-R_ij/1+R_ij  e_(X_n).

Since a_1=0, for any strict partition , we have

    ^(a_1)_(E-E_1) = ∏_i<j1-R_ij/1+R_ij 
    c_(E/E_1) = Q_(X_n).

For any w∈ W_n, we define the symplectic Schubert
polynomial _w(X_n) by 

    _w(X_n):= ∑_v,,e^v_ Q_(X_n)_(-X_n)

where the sum is over all factorizations v = w and strict
partitions  with ℓ(v)+ℓ()=ℓ(w), ∈ S_n, and
||=ℓ(v). Employing computations similar to those in Example
<ref>, we then see that formula (<ref>) is equivalent to
the statement that for any element w∈ W_n, we have
[_w]=_w(X_n) in ^*(_2n/B,).







§ ETA POLYNOMIALS
 




 §.§ Definition and Pieri rule



In the theory of eta polynomials, we must distinguish between the  
case of level zero and that of positive level. Given any strict partition
, the eta polynomial Η^(0)_(u) of level 
0 is defined by 

    Η^(0)_(u) := 2^-ℓ()∏_i<j1-R_ij/1+R_ij  u_.

As a polynomial in the variables u_p, Η^(0)_(u) may have
nonintegral coefficients. However, if we introduce new variables
_p such that u_p=2_p for each p≥ 1, then
Η^(0)_ is a polynomial in the _p with integer
coefficients.


Let ^(0) be the quotient of the polynomial ring 
[_1,_2,…] modulo the ideal of relations

    ^2_p + 2∑_i=1^p-1(-1)^i _p+i_p-i+(-1)^p_2p
    = 0
       for  p ≥ 1.

Then, for any p≥ 0, the Pieri rule

    _p·Η^(0)_(u) = ∑_μ 2^N'(,μ)Η^(0)_μ(u)

holds in ^(0), where the sum is over all strict partitions μ⊃ such that |μ|=||+p and μ/ is a horizontal strip,
and N'(,μ) is one less than the number of connected components of
μ/.




Assume next that k ≥ 1, and let
_1,…,_k-1,_k,_k',_k+1,… be independent
variables related to the variables u_1,u_2,… by the equations

    u_p=
    _p    if p< k,
    _k+_k'    if p=k,
    
    2_p    if p> k.

The eta polynomials Η^(k)_(u) lie in the ring
[_1,…,_k-1,_k,_k',_k+1,…], and are indexed
by typed k-strict partitions .

A typed k-strict partition is a pair consisting of a k-strict
partition  together with an integer in {0,1,2} called the
type of , and denoted (), such that
()=0 if and only if _i≠ k for all i≥ 1. The type
is usually omitted from the notation for the pair (,()). 


For a general typed k-strict
partition , we define the operator

    R^ := ∏_i<j (1-R_ij)∏__i+_j ≥ 2k+j-i
    (1+R_ij)^-1

where the first product is over all pairs i<j and second product is
over pairs i<j such that _i+_j ≥ 2k+j-i.  Let R be any
finite monomial in the operators R_ij which appears in the
expansion of the formal power series R^ in (<ref>).  If
()=0, then set R ⋆ u_ := u_R. Suppose
that ()≠ 0, let r be the least index such that
_r=k, and set 
    :=
    (_1,…,_r-1,_r+1,…,_ℓ)
 for any integer
sequence  of length ℓ. If R involves any factors R_ij
with i=r or j=r, then let R ⋆ u_ := 1/2 u_R. If R has no such factors, then let

    R ⋆ u_ := _k  u_R     if  () = 1, 
    '_k   u_R     if  () = 2.


We define the eta polynomial Η^(k)_(u) of level k by

    Η^(k)_(u) := 2^-ℓ_k() R^⋆ u_.

Here the k-length ℓ_k() of a (typed) k-strict
partition  is the number of parts _i which are strictly
greater than k.  It is easy see that Η_^(k)(u) is a
polynomial in the variables _p and '_k with integer
coefficients. We will write Η_(u) for Η^(k)_(u) when
the level k is understood.


(a) Consider the typed 2-strict partition 
=(3,2,2) with ()=2. Then we have

    Η^(2)_(u)     = 1/21-R_12/1+R_12(1-R_13)(1-R_23) ⋆ u_3,2,2
       = 1/2(1-2R_12+2R_12^2 - 2 R_12^3)(1-R_13-R_23+R_13R_23) 
    ⋆ u_3,2,2
       = _3'_2(_2+_2') - _3^2_1 + _4_3 - _4'_2_1 
    +_6_1-_7.


(b) Let  be a k-strict partition with _i=k≥ 1 for some i, 
and let Η_(u) and Η'_(u) denote the eta polynomials of level
k indexed by  of type 1 and 2, respectively. Then we have

    Η_(u)+ Η'_(u) = 2^-ℓ_k() R^  u_

where R^ denotes the operator (<ref>).


Let ^(k) be the quotient
of polynomial ring  [_1,…,_k-1,_k,_k',_k+1,…]
modulo the ideal of relations

    _p^2+ ∑_i=1^p(-1)^i_p+iu_p-i = 0    for
    p>k, 
    _k'_k+∑_i=1^k (-1)^i_k+i_k-i = 0,

where the u_i obey the equations (<ref>). For every typed
k-strict partition , we define a monomial _ as
follows. If ()≠ 2, then set
_:=__1__2⋯. If ()= 2 then
define _ by the same product formula, but replacing each
occurrence of _k with '_k. The monomials _ and the
polynomials Η_(u) as  runs over all typed
k-strict partitions form two -bases of the graded ring
^(k). The Pieri rule for the products _p·Η_(u) 
holds only modulo the relations (<ref>) and (<ref>);
again we need some further definitions to state it.

We say that the box in row r and column c of a k-strict
partition λ is k'-related to the box in row r' and
column c' if |c-k-1/2|+r = |c'-k-1/2|+r'.  For example, the two
grey boxes in the following partition are k'-related.

    0.65kprshift

For any two k-strict partitions λ and μ, the
relation λμ is defined as in Section
<ref>, but replacing `k-related' by `k'-related'
throughout. The set  of boxes of μ
is defined in the same way, and the integer N'(,μ) is equal to
the number (respectively, one less than the number) of connected
components of , if p≤ k (respectively, if p>k).

If λ and μ are typed k-strict partitions, then we write
λμ if the underlying k-strict partitions
satisfy λμ, with the added condition that
()+(μ)≠ 3.  Let c(,μ) be the number of
columns of μ among the first k which do not have more boxes than
the corresponding column of , and

    d(,μ) := c(,μ)+max((),(μ)).

If p≠ k, then set δ_μ=1. If p=k and 
N'(, μ)>0, then set

    δ_μ=δ'_μ:=1/2,

while if N'(, μ)=0, define

    δ_μ :=  
    1    if d(,μ) is odd, 
    
    0    otherwise    and    δ'_μ = 
    1    if d(,μ) is even, 
    
    0    otherwise.


For any typed k-strict partition  and p≥ 0, we then
have the Pieri rule

    _p ·Η_λ(u) = ∑_μδ_μ 2^N'(,μ) Η_μ(u),

in ^(k), where the sum over all typed k-strict partitions μ 
such that μ. Furthermore, the product
'_k·Η_λ(u) is obtained by replacing δ_λμ with δ'_λμ throughout.





 §.§ Cohomology of Grassmannians
 


Equip the vector space ^2n with the nondegenerate symmetric
bilinear form ( , ) defined by the conditions (e_i,e_j)=0 for
i+j≠ 2n+1 and (e_i,e_2n+1-i)=1 for 1≤ i ≤ n. The
special orthogonal group _2n() is the subgroup of
_2n() consisting of those elements g such that
(gv_1,gv_2)=(v_1,v_2), for every v_1,v_2∈^2n. We say that a
subspace V is isotropic if the restriction of ( , )
to V vanishes identically. Since the form is nondegenerate, we
have (V)≤ n for any isotropic subspace V. For
each i∈ [1,2n], let F_i denote the -linear span of
e_1,…,e_i.

Fix an integer k with 0≤ k ≤ n-1. If k≥ 1, then the
orthogonal Grassmannian =(n-k,2n) parametrizes isotropic
linear subspaces of dimension n-k in ^2n. The group _2n
acts transitively on (n-k,2n), and we have (n-k,2n) =
_2n/P_k, where P_k is a maximal parabolic subgroup of
_2n. When k=0, the locus of maximal isotropic subspaces has
two isomorphic connected components, called the two families,
each of which is a single _2n-orbit.  The orthogonal
Grassmannian (n,2n)=_2n/P_0 parametrizes one of these
components, which we take to be the family containing F_n.

We agree that when k=0, a typed 0-strict partition is the same as
a strict partition, and that all such partitions have type 1. The
Schubert cells ^∘_ in (n-k,2n) are indexed by the
typed k-strict partitions  whose diagrams are contained in an
(n-k)× (n+k-1) rectangle. We have

    X^∘_ := { V ∈|(V ∩
       F_r) = #{j | p_j()≤ r}   ∀  r },

where the strictly increasing index function
{p_j(λ)}_1≤ j ≤ n-k is defined by

    p_j(λ) := n+k+j-λ_j - 
       #{ i<j | _i+_j ≥ 2k+j-i }
     -  
          1    if λ_j > k, or λ_j=k < λ_j-1 and
            n+j+(λ) is odd, 
    
          0    otherwise.

The Schubert variety _ is the closure of the
^∘_, and has codimension || in . 
If [_] denotes the cohomology class of _ in
^2||(,), then we have a group isomorphism

    ^*((n-k,2n),) ≅⊕_ [_].


The varieties _p for 1≤ p ≤ n+k-1, together with _k'
when k≥ 1, are the special Schubert varieties, and their
classes in cohomology are the special Schubert classes.  The
convention here is that _k (respectively, _k') is indexed by
the partition k of type 1 (respectively, type 2). As in the Lie
types A and C, the special Schubert varieties _p and _k' can
be viewed as the locus of all isotropic linear subspaces V which meet
a given isotropic or coisotropic linear subspace nontrivially.


Let Q→(n-k,2n) denote the 
universal quotient vector bundle over , of rank n+k. For k=0,
we have c_p(Q)=2[_p] for all p≥ 1, while for k≥ 1, we have

    c_p(Q)=  [_p]    if p<k, 
    
    [_k]+[_k']    if p=k, 
    
    2[_p]    if p>k

in ^2p((n-k,2n),), in agreement with (<ref>). We can
now state the Giambelli formula

    [_] = Η_(c(Q))

where the polynomial Η_(c(Q)) is obtained from
Η^(k)_(u) by performing the substitutions _p↦ [_p]
and '_k↦ ['_k] for every integer p.

Furthermore, the Pieri rules (<ref>) and (<ref>) hold
in ^*((n-k,2n),). For instance, the latter rule is valid for
k≥ 1 and states that

    [_p]· [_] = ∑_μδ_μ 2^N'(,μ)[_μ]

summed over all typed k-strict partitions 
μ such that μ and the diagram of μ fits 
in an (n-k)× (n+k-1) rectangle. Moreover, the product 
['_k]· [_] is obtained by replacing δ_λμ with δ'_λμ in (<ref>).



For the Grassmannian (5,14) we have n=7 and k=2. Let 
denote the partition (8,7,2,1,1) of type 1. We then have the 
Pieri formulas

    [_2]· [_]     = [_8,7,4,1,1]+[_8,7,3,2,1]+[_8,7,6] 
     
    ['_2]· [_]    = [_8,7,4,1,1]+[_8,7,3,2,1]

where the indexing partitions on the right hand side are all of type 0 or 1.




The ring ^(k) is naturally isomorphic to the stable cohomology
ring

    (_k) = lim_⟵^*((n-k,2n),)

of the orthogonal Grassmannian , where the inverse limit is
defined as in Section <ref>. Under this isomorphism, the variables
_p and '_k map to the special Schubert classes [_p] and
['_k] in the cohomology ring of .






 §.§ Symmetric polynomials



Let b:=(b_1,b_2,…) be a sequence of commuting variables, and
set b_0:=1 and b_p=0 for p<0. Consider the graded ring Γ'
which is the quotient of the ring [b] modulo the ideal
generated by the relations

    b_p^2+2∑_i=1^p-1(-1)^i b_p+ib_p-i+(-1)^p b_2p=0,    for all p≥ 1.

The ring Γ' is isomorphic to ^(0) and to the stable 
cohomology ring

    lim_⟵^*((n,2n),)

of the maximal orthogonal Grassmannian (n,2n), with the variables
b_p mapping to the special Schubert classes [_p]. We regard Γ
as a subring of Γ' using the injection which sends c_p to 
2b_p for all p≥ 1.


The Weyl group W_n for the root system D_n is the
subgroup of W_n consisting of all signed permutations with an even
number of sign changes.  The group W_n is an extension of S_n
by the element s_=s_0s_1s_0, which acts on the right by

    (w_1,w_2,…,w_n)s_=(w_2,w_1,w_3,…,w_n).



Fix n≥ 2 and let X_n:=(x_1,…,x_n). There is a natural action of 
W_n on Γ'[X_n] which extends the 
action of S_n on [X_n], defined as follows. 
The simple reflections s_i for i>0 act
by interchanging x_i and x_i+1 and leaving all the remaining
variables fixed. The reflection s_ maps 
(x_1,x_2) to (-x_2,-x_1), fixes the x_j for j≥ 3, 
and satisfies, for any p≥ 1,

    s_(b_p) :=
    b_p+(x_1+x_2)∑_j=0^p-1(∑_a+b=jx_1^ax_2^b)
    c_p-1-j.

If t is a formal variable which is fixed by s_, then we 
express equation (<ref>) using generating functions as

    s_(∑_p=0^∞ c_pt^p) = 1+x_1t/1-x_1t·1+x_2t/1-x_2t·(∑_p=0^∞ c_pt^p).


For every integer p, define an element ^nb_p of Γ'[X_n] by

    ^nb_p:= 
    e_p(X_n) + 2∑_i=0^p-1 e_i(X_n)b_p-i   if p<n, 
    ∑_i=0^p e_i(X_n)b_p-i   if p≥ n,

let

    ^nb'_n:=∑_i=0^n-1e_i(X_n)b_n-i

and set B^(n):= [^nb_1, …, ^nb_n-1,^nb_n, ^nb'_n,
^nb_n+1, …]. Observe that we have

    ^nc_p=^nb_p    if p<n, 
    ^nb_n+^nb'_n    if p=n, 
    
    2·^nb_p    if p>n

and thus Γ^(n) is a subring of B^(n). 

Let Γ'[X_n]^W_n denote the subring of
W_n-invariants in Γ'[X_n]. Then there is an equality

    Γ'[X_n]^W_n=B^(n)= [^nb_1, …, ^nb_n-1,^nb_n, ^nb'_n,
    ^nb_n+1, …].


The map which sends _p to ^nb_p for every integer p and '_n
to ^nb'_n induces a ring 
isomorphism ^(n)≅ B^(n). We therefore have

    B^(n)=⊕_ Η_(X_n)

where the sum is over all typed n-strict partitions , and the
polynomial Η_(X_n) is obtained from the eta polynomial
Η^(n)_(u) by making the substitutions _p↦^nb_p for
all p and '_n↦^nb'_n. In other words, we have

    Η_(X_n)=Η^(n)_(X_n) := 2^-ℓ_n() R^⋆ (^nc)_.


For each r≥ 2, we embed W_r in W_r+1 by adding
the element r+1 which is fixed by W_r, and set W_∞
:=∪_r W_r.  Let w be a signed permutation in
W_∞.  Define a strict partition μ(w) whose parts are
the absolute values of the negative entries of w minus one, arranged
in decreasing order.  Let the A-code of w be the sequence
γ with γ_i:=#{j>i | w_j<w_i}, define a
partition δ(w) whose parts are the nonzero entries γ_i
arranged in weakly decreasing order, and let ν(w) be the conjugate
of δ(w). The shape of w is defined to be the partition
(w):=μ(w)+ν(w).



(a) An element w of W_∞ is n-Grassmannian if
  ℓ(ws_i)>ℓ(w) for all i≠ n. The type of an
  n-Grassmannian element w is 0 if |w_1|=1, and 1 (respectively,
  2) if w_1>1 (respectively, if w_1<-1). There is a type
  preserving bijection between the n-Grassmannian elements of
  W_∞ and typed n-strict partitions, given as follows.
  If the element w corresponds to the typed n-strict partition
  , then for each j≥ 1, we have

    _i= 
    n-1+|w_n+i|    if w_n+i<0, 
    #{r≤ n  :  |w_r|> w_n+i}   if w_n+i>0.

The shape (w) of w agrees with the typed n-strict partition 
associated to w, if w has type 0 or 1. However, this may fail if
w_1<-1, for instance the 2-Grassmannian element 
v:=35124 in W_5 is associated to the
typed partition of shape (2,2,1), while (v)=(3,1,1).

(b) The longest element of W_n is given by

    w_0={[         (1,…,n) if  n  is even,;       (1,2,…,n)  if  n  is odd. ].

Then μ(w_0)=δ_n-1, ν(w_0)=
δ_n-1, and (w_0)=2δ_n-1=(2n-2,2n-4,…,
2).


Let w be an n-Grassmannian element of W_∞ with corresponding 
partition λ(w), and let w_0 be the longest element of W_n.
Define

    ^ν(ww_0)P_(ww_0) := 2^-r ∏_i<j1-R_ij/1+R_ij ^ν(ww_0)c_(ww_0),

where r is the length of the partition (ww_0).
The alternating operator ” on Γ'[X_n] is given by

    ”(f):=∑_w∈W_n (-1)^ℓ(w)w(f).

We then have

    Η_(w)(X_n) 
    =(-1)^n(n-1)/2· 2^n-1. 
    ”(^ν(ww_0)P_(ww_0))”(x^(w_0))

in Γ'[X_n].





 §.§ Algebraic combinatorics



Let k be a nonnegative integer. In this section, we will define the 
formal power series Η^(k)_(Z ;X_k) which are the analogues in Lie
type D of the power series ^(k)_(Z ;X_k) discussed in Section 
<ref>. Their definition is easy when k=0: in this case, the index
 is a strict partition and we have

    Η^(0)_(Z) := 2^-ℓ()^(0)_(Z) = P_(Z),

where P_(Z) is a Schur P-function. The map which sends _j
to P_j:=P_j(Z)=Q_j(Z)/2 for every positive integer j gives an
isomorphism between the ring ^(0)≅Γ' defined in Section
<ref> and the ring [P_1,P_2,…] of Schur
P-functions. Since Η^(0)_(Z) is a scalar multiple of
^(0)_(Z)=Q_(Z), we immediately obtain a tableau formula
for Η^(0)_(Z) from the tableau formula (<ref>) for the
Schur Q-functions.

Assume next that k≥ 1. Set

    η_r=η_r(Z ;X_k) := 
    e_r(X_k) + 2∑_i=0^r-1 P_r-i(Z)e_i(X_k)    if r<k, 
    ∑_i=0^r P_r-i(Z) e_i(X_k)    if r≥ k

and 
    η'_k=η'_k(Z ;X_k) =  ∑_i=0^k-1 P_k-i(Z) e_i(X_k).
 
For any r≥ 0, if _r is defined by equation (<ref>),
then we have

    _r=
    η_r    if r< k,
    η_k+η_k'    if r=k,
    
    2η_r    if r> k.


By definition, for every typed k-strict partition ,
the eta polynomial Η_(Z ;X_k) is obtained from
Η^(k)_(u) by making the substitutions _r↦η_r
for every integer r and '_k↦η'_k. In other words, we have

    Η_(Z ;X_k):= 2^-ℓ_k() R^⋆_.

We proceed to give a tableau formula for Η_(Z ;X_k) which is 
analogous to the formulas (<ref>) and (<ref>).


Let  and μ be k-strict partitions with μ⊂,
and choose any p> ||+2k-1. If
||=|μ|+r and (p+r,μ), then we 
say that /μ is a k'-horizontal strip.  
We call a box in row r and column c of a Young diagram 
a left box if c ≤ k and a right box if c>k.
If μ⊂ are two k-strict
partitions such that /μ is a k'-horizontal strip,
we define _0=μ_0=+∞ and agree that the
diagrams of  and μ include all boxes [0,c] in row zero.
We let  denote the set of right boxes of μ
(including boxes in row zero) which are bottom boxes of  in their
column and are not (k-1)-related to a left box of
/μ. 


If  and μ are typed k-strict partitions with
μ⊂, we say that /μ is a
typed k'-horizontal strip if the underlying k-strict
partitions are such that /μ is a k'-horizontal strip and in
addition ()+(μ)≠ 3. In this case we let
n(/μ) denote  the number of connected components of
 minus one.



Suppose that  is any typed k-strict partition.  Let P'
denote the ordered alphabet
{1<2<⋯<k<1,1^∘<2,2^∘<⋯}.  We say 
that the symbols 1,…,k are marked, while the
rest are unmarked. A typed k'-tableau T of shape
/μ is a sequence of typed k-strict partitions

    μ = ^0⊂^1⊂⋯⊂^r=

such that ^i/^i-1 is a typed k'-horizontal strip for
1≤ i≤ r.  We represent T by a filling of the boxes in
/μ with unmarked elements of P' which is weakly
increasing along each row and down each column, such that for each
i, the boxes in T with entry i or i^∘ form the skew
diagram ^i/^i-1, and we use i (resp. i^∘) if and
only if (^i)≠ 2 (resp. (^i)=2), for every
i≥ 1.  For any typed k'-tableau T we define n(T)=∑_i
n(^i/^i-1) and let c(T)=(r_1, r_2, …) be the content
vector of T, so that r_i denotes the number of times that i or
i^∘ appears in T, for each i≥ 1.

A typed k'-bitableau U of shape  is a filling of the
boxes in the diagram of  with elements of P' which is
weakly increasing along each row and down each column, such that the
unmarked entries form a typed k'-tableau T of shape /μ with
(μ)≠ 2, and the marked entries are a filling of μ
which is strictly increasing along each row. We define

    n(U)=n(T)   and  
    (zx)^c(U)= z^c(T) ∏_j=1^k x_j^m_j

where m_j denotes the number of times that j appears in U.
For any typed k-strict partition , we then have the tableau formula

    Η_(Z ;X_k) = ∑_U 2^n(U)(zx)^c(U)

where the sum is over all typed k'-bitableaux U of shape . 
Using the tableau formula (<ref>), we can rewrite
equation (<ref>) as 

    Η_(Z ;X_k) = ∑_μ⊂∑_T 2^n(T)z^c(T) s_μ(X_k)

with the sums over all partitions μ⊂ and typed k'-tableaux T
of shape /μ, respectively.



Suppose that k=1, and for any r≥ 1, let Η_1^r(Z ;X_1)
and Η'_1^r(Z ;X_1) denote the eta polynomials indexed by 1^r of
type 1 and 2, respectively. Then for any integer r≥ 0, we
deduce from equation (<ref>) that

    Η_r(Z ;X_1)    = P_r(Z)+P_r-1(Z)x_1,
    Η_1^r(Z ;X_1)    = P_r(Z) + 2 P_r-1(Z)x_1 + ⋯ + 
    2 P_1(Z)x_1^r-1+x_1^r,  and
    Η'_1^r(Z ;X_1)    = P_r(Z).





Consider the set _ :={,1,…} whose members index the
simple reflections in W_∞. These elements generate the
group W_∞ and are used to define reduced words and the
length of signed permutations as in Section <ref>.  The nilCoxeter algebra _n of the group W_n is the free
associative algebra with unit generated by the elements
ξ_,ξ_1,…,ξ_n-1, modulo the relations

    [                      ξ_i^2                          =                         0,                      i∈_ ;;                      ξ_ξ_1                          =                    ξ_1 ξ_,;                   ξ_ξ_2 ξ_                          =                 ξ_2 ξ_ξ_2,;                ξ_iξ_i+1ξ_i                          =             ξ_i+1ξ_iξ_i+1,                      i>0 ;;                     ξ_iξ_j                          =                    ξ_jξ_i, j> i+1,  and (i,j) ≠ (,2). ]

As in the previous sections, for any w∈W_n, choose a reduced
word a_1⋯ a_ℓ for w and define ξ_w := ξ_a_1…ξ_a_ℓ. The ξ_w form a free -basis of _n, and
we denote by ⟨ζ,w⟩ the coefficient of ξ_w in
the expansion of the element ζ∈_n.

Let t be an independent variable, define

    D(t) := (1+t ξ_n-1)⋯ (1+t ξ_2)(1+t ξ_1)(1+t ξ_)
    (1+t ξ_2)⋯ (1+t ξ_n-1),

and let D(Z)=D(z_1)D(z_2)⋯.
Choose an integer k with 0≤
k<n.  For any w∈ W_n, the (restricted) type D mixed Stanley
  function I_w(Z ;X_k) is defined by

    I_w(Z ;X_k) := ⟨ D(Z)A(X_k), w⟩.

The power series I_w(Z ;X_k) is symmetric in the Z and X_k
variables, separately, and has nonnegative integer coefficients.

When w is the k-Grassmannian element associated to a typed
k-strict partition , then we have

    I_w(Z ;X_k)=Η_(Z ;X_k).

We say that an element w=(w_1,…,w_n) has a descent at
position i∈_ if ℓ(ws_i)<ℓ(w).  If k≥ 2, we say
that w is increasing up to k if it has no descents less than
k; this means that |w_1|< w_2<⋯ < w_k. By convention we agree
that every element of W_∞ is increasing up to  and
also increasing up to 1.  We can now state the following
generalization of equality (<ref>): for any element w∈W_n
which is increasing up to k, we have

    I_w(Z ;X_k) = ∑_d^w_ Η_(Z ;X_k)

summed over typed k-strict partitions  with ||=ℓ(w).
The integers d^w_ are nonnegative, in other words, the function
I_w(Z ;X_k) is eta positive when w is increasing up to
k. A combinatorial interpretation for these coefficients is
provided below.

For any w∈W_∞ which is increasing up to k, we
construct the k-transition tree T^k(w) with nodes given by
elements of W_∞ and root w in a manner parallel to  
Section <ref>. Let r be the largest descent of w. If w=1, or k≠ 1
and r=k, or k=1 and r∈{,1}, then set T^k(w)
:={w}. Otherwise, let s := max(i>r | w_i < w_r) and define
Φ(w) := Φ_1(w)∪Φ_2(w), where

    Φ_1(w) := {wt_rst_ir | 1≤ i < r   and  ℓ(wt_rst_ir) = ℓ(w) }, 
    Φ_2(w) := 
    {wt_rst_ir | i≠ r   and ℓ(wt_rst_ir) = ℓ(w) }.

To define T^k(w), we join w by an edge to each v∈Φ(w), and attach to each v∈Φ(w) its tree
T^k(v). Then T^k(w) is a finite tree called the k-transition tree of w, and its leaves are all k-Grassmannian
elements of W_∞.  The coefficient d^w_ in (<ref>)
is equal to the number of leaves of T^k(w) of shape .





 §.§ Cohomology of flag manifolds


Let {e_1,…,e_2n} denote the standard orthogonal basis of
E:=^2n and let F_i=⟨ e_1,…, e_i⟩ be the
subspace spanned by the first i vectors of this basis, as in Section
<ref>.  The group G=_2n acts on the space of all complete
isotropic flags in E with two orbits, determined by the family of
the maximal isotropic subspace in a given flag.  The stabilizer of the
flag F_ is a Borel subgroup B of G. Let T⊂ B denote
the maximal torus of diagonal matrices in G, and the Weyl group
W=N_G(T)/T≅W_n.


The parabolic subgroups P of _2n with P⊃ B
correspond to sequences a_1<⋯ < a_p of elements of _
with a_p<n. For any such P, the manifold :=_2n/P
parametrizes partial flags of subspaces

    E_ :  0 ⊂ E_p ⊂⋯⊂ E_1 ⊂ E=^2n

with E_1 isotropic, (E_r) = n-a_r for each r∈ [1,p], and
E_1 in a given family if a_1=. The associated parabolic
subgroup W_P of W_n is generated by the simple reflections
s_i for i∉{a_1,…, a_p}.  As usual, E_r for r∈
[1,p] and E will also denote the corresponding tautological vector
bundles over .

There is a canonical presentation of the cohomology ring of _2n/B,
which gives geometric significance to the variables which appear in
Section <ref>. Let ^(n) denote the ideal of 
Γ'[X_n]_:=Γ'[X_n]⊗_
generated by the homogeneous elements of positive degree in
B^(n), so that ^(n)=⟨^nb_n',^nb_1,^nb_2,…⟩. We then have ring isomorphism

    ^*(_2n/B,) ≅Γ'[X_n]_/^(n)

which maps each variable b_p to c_p(E/E_n)/2 and x_i to
c_1(E_n+1-i/E_n-i) for 1≤ i ≤ n. Furthermore, for any
parabolic subgroup P of _2n, the projection map _2n/B→_2n/P induces an injection ^*(_2n/P)↪^*(_2n/B) on cohomology rings, and we have

    ^*(,) ≅Γ'[X_n]_^W_P/^(n)_P,

where Γ'[X_n]_^W_P denotes the W_P-invariant subring of
Γ'[X_n]_, and _P^(n) is the ideal of Γ'[X_n]_^W_P 
generated by ^nb_n',^nb_1,^nb_2,…

We have a decomposition

    _2n = ⋃_w ∈ W^PBw P

where 

    W^P := {w∈W_n | ℓ(ws_i) = ℓ(w)+1, ∀  i ∉{a_1,…, a_p},  i<n}

is the set of minimal length W_P-coset representatives in W_n.
For each w ∈ W^P, the B-orbit of wP in _2n/P is the
Schubert cell ^∘_w:=Bw P/P. The Schubert variety
_w is the closure of ^∘_w in . Then _w:=_w_0w
has codimension ℓ(w) in _2n/P, and its cohomology class 
[_w] is a Schubert class. We have an isomorphism of
abelian groups

    ^*(,) ≅⊕_w∈ W^P[_w]

which generalizes (<ref>). 

Recall that E_r for r∈ [1,p] and E denote the tautological and
trivial vector bundles over , of rank n-a_r and 2n,
respectively. For any w∈ W^P, we then have

    [_w] = ∑_ g^w_ Η^(a_1)_^1(E-E_1) s_^2(E_1-E_2)⋯ s_^p(E_p-1-E_p)

in ^*(,), where the sum is over all sequences of
partitions =(^1,…,^p) with ^1 being 
typed a_1-strict, and the 
coefficients g^w_ are given by 

    g^w_ := ∑_u_1⋯ u_p = w
    d_^1^u_1c_^2^u_2⋯ c_^p^u_p

summed over all factorizations u_1⋯ u_p= w such that
ℓ(u_1)+⋯ + ℓ(u_p)=ℓ(w), u_j∈ S_∞ for j≥
2, and u_j(i)=i for all j>1 and i ≤ a_j-1.  The
nonnegative integers d^u_1_^1 and c^u_i_^i which
appear in the summands in (<ref>) are the same as the ones in
equations (<ref>) and (<ref>), respectively.  When p=1, the
partial flag manifold  is the orthogonal Grassmannian
(n-a_1,2n), and formula (<ref>) specializes to equation
(<ref>). In general, the polynomial
Η^(a_1)_^1(E-E_1) in (<ref>) is defined by pulling
back the polynomial Η^(a_1)_^1(c(E/E_1)) under the natural
projection map E_↦ E_1 from  to (n-a_1,2n).




Let P=B be the Borel subgroup, so that the flag manifold
_2n/B parametrizes flags of subspaces 0⊂ E_n ⊂⋯⊂ E_1 ⊂ E=^2n with E_1 isotropic in a given
family, and (E_i) = n+1-i for each i∈ [1,n]. For any strict
partition , define the P-polynomial
P_(X_n):=2^-ℓ() Q_(X_n). Since a_1=0, we
have Η^(a_1)_(E-E_1) = P_(X_n), where we set
x_i:=-c_1(E_i/E_i+1) for each i. For any element w∈W_n, define the orthogonal Schubert polynomial
_w(X_n) by

    _w(X_n):= ∑_v,,d^v_ P_(X_n)_(-X_n)

where the sum is over all factorizations v = w and strict
partitions  such that ℓ(v)+ℓ()=ℓ(w), ∈ S_n,
and ||=ℓ(v). Working as in Example <ref>, one can show
that formula (<ref>) is equivalent to the statement that for
any element w∈W_n, we have [_w]=_w(X_n) in
^*(_2n/B,).




§ HISTORICAL NOTES AND REFERENCES





 §.§ Schur polynomials


The Schur polynomials s_(X_n) were first defined in the early
19th century by Cauchy <cit.> using formula (<ref>), as a
quotient of two alternant determinants. These polynomials were studied
further by Jacobi <cit.> and his student Trudi <cit.>, who
established the determinantal formula (<ref>). The dual
identity (<ref>) was proved by Nägelsbach <cit.>. The
reformulation of the Jacobi-Trudi identity using raising operators
(<ref>) originates in Young's work on the representation
theory of the symmetric group; see <cit.> and compare
with <cit.>. 

The Pieri and Giambelli formulas (<ref>) and (<ref>) for
Grassmannians were proved in <cit.> and <cit.>, respectively.
In his thesis <cit.>, Schur showed that the polynomials
s_(X_n) can be viewed as the characters of the irreducible
polynomial representations of the general linear group _n. From
the perspective of representation theory, equation (<ref>)
is a special case of the Weyl character formula. Finally, the tableau
formula (<ref>) for Schur polynomials was established by
Littlewood <cit.>, more than 100 years after Cauchy's definition
appeared. For an approach to the theory of Schur polynomials starting
from the raising operator definition (<ref>), which
includes the above topics and more, see <cit.>.

The code γ() of a permutation  is the Lehmer code used
in computer science, which was known in the 19th century (see for
example <cit.>). The shape () of a permutation was defined
in <cit.>. The Stanley symmetric functions G_(X) were
introduced in <cit.>; in the notation of op. cit., the function
G_^-1(X) is assigned to . Stanley's conjecture that the
coefficients c^_ in (<ref>) are nonnegative was proved
independently by Edelman and Greene <cit.> and Lascoux and
Schützenberger <cit.>; the latter authors introduced the
transition trees of Section <ref>. The definition of G_(X)
using the nilCoxeter algebra is due to Fomin and Stanley <cit.>.

The presentations (<ref>) and (<ref>) of the cohomology
of type A flag manifolds are due to Borel <cit.>. Formula
(<ref>) was proved in <cit.>, and grew out of a study of
the Schubert polynomials _(X_n) of Lascoux and Schützenberger
<cit.> and their relation to the quiver polynomials of Buch
and Fulton <cit.>. The definition of _(X_n) given in
formula (<ref>) is found in <cit.>.




 §.§ Theta polynomials


In order to clarify the relevant history, the arXiv announcement years
are listed for the main papers below, since their publication dates in
journals have little to do with when the work was completed. The story
begins with the companion papers <cit.> (arXiv:2008) and
<cit.> (arXiv:2008) which studied Schubert calculus on non
maximal isotropic Grassmannians.  The first paper <cit.> proved
the Pieri rule (<ref>), while <cit.> dealt with the
Giambelli formula (<ref>) and theta polynomials.

The parametrization of the Schubert classes on (n-k,2n) by
k-strict partitions was introduced in <cit.>. As explained in
op. cit., although the k-strict partitions are not really needed
there, they are a key ingredient of its companion paper <cit.>,
and in related works such as <cit.> (arXiv:2008). 

The paper <cit.> was the first to realize that Young's raising
operators play an essential role in geometry, in the Giambelli type
formulas for isotropic Grassmannians, and to employ them in their
proofs. Before <cit.>, these operators made occasional
appearances, notably in the theory of Hall-Littlewood polynomials (see
e.g. <cit.>), but they were rarely used, even in
representation theory and combinatorics. The solution to the Giambelli
problem for the usual (type A), Lagrangian, and maximal orthogonal
Grassmannians found in <cit.> and <cit.>, respectively, employed
the older language of Jacobi-Trudi determinants (in Lie type A) and
Schur Pfaffians (in types B, C, and D), which goes back to <cit.> and
<cit.>.

In the initial version of <cit.>, the theta polynomials were
expressed as the formal power series in (<ref>), although
the intention in op. cit. (which justifies the term `polynomial')
was to regard the _ as Giambelli polynomials in the
_p. The definition (<ref>) of the theta polynomials
_(u) in independent variables u_i was first given explicitly
in <cit.> (arXiv:2009).


The ring Γ of Section <ref> is isomorphic to the ring of
Schur Q-functions (see Example <ref>, <cit.>, and
<cit.>), whose elements are symmetric formal power series in
Z:=(z_1,z_2,…). Using the latter notation, the action of
W_∞ on the ring Γ[X] was studied by Billey and Haiman in
<cit.>.  The same authors obtained a natural -basis
of Γ[X] consisting of type C Schubert polynomials (actually
power series) C_w(X), for w∈ W_∞. When w is an
n-Grassmannian element of W_∞ of shape given by the
n-strict partition , then C_w(X) is equal to _(X_n)
in Γ[X], so modulo the defining relations in Γ (see
<cit.>).

It was shown in <cit.> that the theta polynomials _(X_n)
of level n are symmetric for the W_n-action on Γ[X_n], and
form a -basis for the W_n-invariants there, which is the content
of (<ref>) and (<ref>). The paper <cit.> goes on to
define the shape (w) of a signed permutation w, proves equation
(<ref>), and also obtains the presentations (<ref>) and
(<ref>).  We note that multi-Schur Pfaffians such as
(<ref>) first appeared in the work of Kazarian <cit.>,
resurfaced in <cit.> (arXiv:2008), and were subsequently used in
the degeneracy locus formulas of <cit.>.

The tableau formula (<ref>) for _(Z ;X_k) was
established in <cit.>. The fact that the left weak Bruhat
  order on the k-Grassmannian elements of W_∞ respects the
inclusion relation ⊂μ of k-strict Young diagrams is
what makes such a formula possible. This was pointed out in
<cit.>. Example <ref> is well known in the 
combinatorial theory of Schur Q-functions; see for instance 
<cit.>.

The type C mixed Stanley functions and k-transition trees were
defined and equation (<ref>) was proved in <cit.>. These
constructions build on earlier results of Billey and Haiman <cit.> and Fomin and Kirillov <cit.>, which had studied the k=0
case. The paper <cit.> also proves formula (<ref>), which
gives an intrinsic solution to the Giambelli problem of representing
the Schubert classes in the cohomology of _2n/P, for any
parabolic subgroup P of _2n. 

The Q-polynomials Q_(X_n) in Example <ref> are
due to Pragacz and Ratasjki <cit.>. The symplectic Schubert
polynomials _w(X_n) in (<ref>) were defined in <cit.>
(arXiv:2008), and are a geometrization of the Billey-Haiman type
C Schubert polynomials. For more on this history, we refer the reader
to <cit.>.




 §.§ Eta polynomials


The paper <cit.> introduced typed k-strict partitions and
proved the Pieri rule (<ref>) for non maximal even orthogonal
Grassmannians. Young's raising operators were used in <cit.>
(arXiv:2011) to define eta polynomials and to prove equation
(<ref>), which solves the Giambelli problem for the same
spaces. The work <cit.> extended this to address the analogous
question for all partial orthogonal flag manifolds, which gives
equation (<ref>). See <cit.> for a detailed exposition,
which includes the version of this result which holds in the more
general setting of degeneracy loci of vector bundles for the classical
Lie groups.

The ring Γ' of Section <ref> is isomorphic to the ring
[P_1,P_2,…] of Schur P-functions. The action of
W_∞ on the ring Γ'[X] and the induced divided
difference operators there were studied by Billey and Haiman
<cit.>, who defined a -basis of Γ'[X] consisting of type
D Schubert polynomials D_w(X) for w∈W_∞, compatible
with these operators.  According to <cit.>, when w
is an n-Grassmannian element of W_∞ associated to the
typed n-strict partition , then D_w(X) is equal to
Η_(X_n) in Γ'[X]. Again it is important to note that
this equality takes place in a ring with relations, which come from
the subring Γ'.

The fact that the eta polynomials of level n are symmetric for the
W_n-action on Γ'[X_n] and provide a -basis for the
Weyl group invariants there, which is the content of (<ref>) and
(<ref>), was explained in <cit.>.  The paper <cit.> also
defines the shape (w) of an element w of W_∞,
proves equation (<ref>), and moreover obtains the
presentations (<ref>) and (<ref>).


The tableau formula (<ref>) for the eta polynomials
Η_(Z ;X_k) was established in <cit.> (arXiv:2011). The
type D mixed Stanley functions and k-transition trees were defined
and equation (<ref>) was proved in <cit.>.  This used the even
orthogonal (type D) version of the nilCoxeter algebra approach of
<cit.> to type A and B Stanley symmetric functions, which is
found in <cit.>. The P-polynomials P_(X_n) and
orthogonal Schubert polynomials _w(X_n) of Example <ref>
were defined in <cit.> and <cit.> (arXiv:2009), respectively.



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[T6]T6 H. Tamvakis : 
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[T7]T7 H. Tamvakis : 
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