Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure Characters, and Nilpotent Varieties.
Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain higher level Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.
Introduction
In [9, Theorem 5.2] it was shown that the characters of certain level one Demazure
modules of type A (1)
decomposed as linear combinations of irreducible
characters of type An\Gamma1 , have coefficients given by Kostka-Foulkes polynomials in
the variable is the null root. The key steps in the proof are that
1. The Demazure crystals are isomorphic to tensor products of classical b
sl n crystals
indexed by fundamental weights [10].
2. The generating function over these crystals by weight and energy function is
equal to the generating function over column-strict Young tableaux by weight
and charge [22].
3. The Kostka-Foulkes polynomial is the coefficient of an irreducible sl n -character
in the tableau generating function [1] [17].
The main result of this paper is that for a Demazure module of arbitrary level
whose lowest weight is a multiple of one of those in [9], the corresponding coefficient
polynomial is the Poincar'e polynomial of an isotypic component of the coordinate
ring of the closure of the conjugacy class of a nilpotent matrix. The Poincar'e
polynomial is the q-analogue of the multiplicity of an irreducible gl(n)-module in a
tensor product of irreducible gl(n)-modules in which each factor has highest weight
given by a rectangular (that is, a multiple of a fundamental) weight. These poly-
nomials, which possess many properties generalizing those of the Kostka-Foulkes,
have been studied extensively using algebro-geometric and combinatorial methods
[8] [13] [24] [25] [30] [31].
The connection between the Demazure modules and the nilpotent adjoint orbit
closures can be explained as follows. Let X - be the Zariski closure of the conjugacy
class of the nilpotent Jordan matrix with block sizes given by the transpose partition
- t of -, that is,
Lusztig gave an embedding of the variety X - as an open dense subset of a P -
stable Schubert variety Y - in c
is the parabolic subgroup
given by "omitting the reflection r 0 " [19]. The desired level l Demazure module,
viewed as an sl n -module, is isomorphic to the dual of the space of global sections
L\Omega l), where L   0 is the restriction to Y - of the homogeneous line bundle on
c
SLn=P affording the fundamental weight   0 .
The proof of the main result entails generalizations of the three steps in the proof
of [9, Theorem 5.2]. First, the methods of [9] may used to show that the Demazure
crystal is isomorphic to a classical b
sl crystal B that is a tensor product of crystals
of the form B k;l (notation as in [7]). We call B k;l a rectangular crystal since it
is indexed by the weight l  k that corresponds to the rectangular partition with k
rows and l columns.
Second, it is shown that the crystal B is indexed by sequences of Young tableaux
of rectangular shape equipped with a generalized charge map. In particular, we
give explicit descriptions in terms of tableaux and the Robinson-Schensted-Knuth
correspondence, of
ffl The zero-th crystal raising operator ee 0 acting on B, which involves the generalized
cyclage operators of [24] on LR tableaux and a promotion operator
on column-strict tableaux.
ffl The combinatorial R-matrices on a tensor product of the form B k1 ;l
which are given by a combination of the generalized automorphism of conjugation
[24] and the energy function.
ffl The energy function, which equals the generalized charge of [13] [24].
Moreover, it is shown that every generalized cocyclage relation [24] on LR tableaux
may be realized by ee 0 . The formula for the corresponding change in the energy
function by ee 0 was known [22] in the case that all rectangles are single rows or all
are single columns.
Third, it must be shown that the tableau formula coincides with the Poincar'e
polynomial. This was accomplished in [24], where it is shown that the tableau
formula satisfies a defining recurrence of Weyman [31] [32] for the Poincar'e polynomials
that is closely related to Morris' recurrence for Kostka-Foulkes polynomials
[21].
As an application of the formula for the energy function, we give a very simple
proof of a monotonicity property for the Poincar'e polynomials (conjectured by A. N.
Kirillov) that extends the monotonicity property of the Kostka-Foulkes polynomials
that was proved by Han [4].
Thanks to M. Okado for pointing out the reference [26] which has considerable
overlap with this paper and [24].
2. Notation and statement of main result
2.1. Quantized universal enveloping algebras. For this paper we only require
the following three algebras: U q (sl n
sl n ).
Let us recall some definitions for quantized universal enveloping algebras taken
from [5] and [6]. Consider the following data: a finitely generated Z-module P
(weight lattice), a set I (index set for Dynkin diagram), elements
roots) and f h (basic coroots) such that
is a generalized Cartan matrix, and a symmetric form (\Delta; \Delta) : P \Theta P ! Q such that
Q\Omega P .
This given, let U q ( b
be the quantized universal enveloping algebra, the Q(q)-
algebra with generators fe and relations as in [6,
Section 2].
For U q ( b
1g be the index set for the Dynkin diagram,
the Cartan matrix of type A (1)
(fundamental weights) and let P   have dual basis fh
Define the elements
so that (hh i ; ff j i) i;j2I is the Cartan matrix of type A (1)
the symmetric Q-valued form (\Delta; \Delta) by (ff
the quantized universal enveloping algebra for
this data is U q ( b
sl
be the dominant weights. For
be the irreducible integrable highest weight U q ( b
sl n )-module of
highest weight  , B( ) its crystal graph, and u   2 B( ) the highest weight vector.
For U 0
sl n ), let I and (a ij ) be as above, but instead of P use the "classical
weight lattice" P cl
i2I Z  i (where by abuse of notation the image of
in P cl is also denoted   i ). The basic coroots a Z-basis of P
cl .
The basic roots are fff denotes the image of ff i in P cl for
Note that the basic roots are linearly dependent. The pairing and symmetric form
are induced by those above. The algebra for this data is denoted U 0
may
be viewed as a subalgebra of U q ( b
its generators are a subset of those of
U
its relations map to relations. Let P
cl
the U q ( b
sl n)-module V ( ) is a U 0
sl n)-module by restriction and weights are taken
modulo ffi .
For U q (sl n ), let I be the index set for the Dynkin diagram,
the restriction of the above Cartan matrix to J \Theta J , and P cl = P cl =Z  0
the weight lattice. The basic coroots
cl Jg. form a Z-basis of
cl
. The basic roots are fff cl Jg. The algebra for this data
is U q (sl n ), which can be viewed as a subalgebra of U 0
sl
cl
be the dominant integral weights. For
cl let V - be the irreducible U q (sl n )-
module of highest weight -, and B - its crystal graph.
Denote by W the Weyl group of the algebra b
sl n and by W that of sl n . W is the
subgroup of automorphisms of P generated by the simple reflections fr
where
cl . W acts faithfully on the affine subspace
cl . For for the map X ! X given by translation by -. Let
cl . Then the action of r i on X for i 2 J is given by
is the reflection through the hyperplane orthogonal to '. Then W
the element - 2 Q acts by - .
For the Demazure module of lowest weight w  is defined
by Vw ( is a generator of the (one dimensional) extremal
weight space in V ( ) of weight w  and U q (b) is the subalgebra of U q ( b
by the e i and h 2 P   .
2.2. Main result. Let - be a partition of n. The coordinate ring C [X - ] of the
nilpotent adjoint orbit closure X - has a graded sl n -action induced by matrix conjugation
on X - . For
cl , define the Poincar'e polynomial of the -th isotypic
component of C [X - ], by
d-0
where C [X - ] d is the homogeneous component of degree d.
Partitions with at most n parts are projected to dominant integral weights of
sl n by - 7! wt sl (-), where
wt sl
cl
Remark 1. Warning: this is not the Kostka polynomial, but a generalization;
see [31]. In the special case that - is a partition of n with at most n parts then
which is a renormalization of the Kostka-Foulkes polynomial
with indices - t and -.
Theorem 2. Let l be a positive integer, - a partition of n, and w - 2 W the
translation by the antidominant weight \Gammaw 0 wt sl (- t cl , where w 0 is the longest
element of W . Then
e \Gammal  0 chVw- (l  0
K wt sl (- (q) chV wt sl (-)
runs over the partitions of the multiple ln of n with at most n parts.
3. Crystal structure on tensor products of rectangles
The goal of this section is to give explicit descriptions of the classical b
sl n crystal
structure on tensor products of rectangular crystals and their energy functions.
This is accomplished by translating the theory of sl n crystals and classical b
crystals in [6] [7] [11] [22] into the language of Young tableaux and the Robinson-
Schensted-Knuth (RSK) correspondence.
3.1. Crystals. This section reviews the definition of a weighted crystal [6] and
gives the convention used here for the tensor product of crystals.
A P-weighted I-crystal is a a weighted I-colored directed graph B, that is, a set
equipped with a weight function directed edges colored by the set
I, satisfying the following properties.
(C1) There are no multiple edges; that is, for each i 2 I and b; b there is at
most one edge colored i from b to b 0 .
If such an edge exists, this is denoted b
equivalently
of the notation of a function B ! B. It is said that e
f i (b) is defined or equivalently
that ee i (b 0 ) is defined, if the edge exists.
i (b) is definedg
i (b) is defined.g
f i (b) is defined then wt( e
Equivalently, wt(ee i
is a P-weighted I-crystal for 1 - j - m, the Cartesian product Bm \Theta \Delta \Delta \Delta\ThetaB 1
can be given a crystal structure as follows; this crystal is denoted
The convention used here is opposite that in much of the literature but is convenient
for the tableau combinatorics used later. Let b
The weight function on B is given by
The root operators e
f i and the functions OE i are defined by the "signature rule".
Given construct a biword (sequence of pairs of letters) consisting
of OE i (b j ) copies of the biletter
copies of the biletter
\Delta for all j,
sorted in weakly increasing order by the order
This biword is now repeatedly reduced by removing adjacent biletters whose lower
letters are +\Gamma in that order. If are viewed as left and right parentheses
then this removes matching pairs of parentheses. At the end one obtains a biword
whose lower word has the t. If s ? 0
be the upper letter corresponding to the rightmost
leftmost +) in the reduced biword, and define
e
\Delta\Omega b 1+j
\Gamma\Omega e
respectively,
e
\Delta\Omega b
1+j+\Omega ee i (b j+
\Delta\Omega
A morphism of P-weighted I-crystals is a map g that preserves weights
and satisfies g( e
I and b 2 B, that is, if e
f i (b) is defined
then e
is, and the above equality holds.
It is easily verified that the P-weighted I-crystals form a tensor category.
We only require the following kinds of crystals.
1. The crystal graphs of integrable U q ( b
sl n )-modules are P-weighted I-crystals
and are called b
sl n -crystals.
2. The crystal graphs of U 0
sl n)-modules that are either integrable or are finite-dimensional
and have a weight space decomposition, are P cl -weighted I-
crystals and are called classical b
sl n -crystals.
3. The crystal graphs of integrable U q (sl n )-modules are P cl -weighted J-crystals
and are called sl n -crystals.
3.2. Crystal reflection operator and Weyl group action. Let B be an sl n
crystal and i 2 J . Write
e
e
e
e \Gammap
The Weyl group W acts on B by r i er i (b) for i 2 J . It is obvious that er i is
an involution and that e r i and e r j commute for but not at all obvious
6 MARK SHIMOZONO
that the e
r i satisfy the other braid relation. A combinatorial proof of this fact is
indicated in [18] for the action of W on an irreducible sl n crystal.
3.3. Irreducible sl n crystals. Let - 1
be a partition
of length at most n. Let V - be the irreducible U q (sl n )-module of highest weight
wt sl (-) and B - its crystal. In [11] the structure of the sl n crystal B - is determined
explicitly. The crystal B - may be indexed by the set CST(-) of column-strict
tableaux of shape - with entries in the set ng. The combinatorial
construction yielding the action of the crystal operators e e i and e
already known. In a 1938 paper, in the course of proving
the Littlewood-Richardson rule, G. de B. Robinson gave a form of the Robinson-
Schensted-Knuth (RSK) correspondence which is defined by giving the value of the
map on sl n -highest weight vectors and then extending it by via canonical sequences
of raising operators e
f i [23, Section 5]; see also [20, I.9] where Robinson's proof is
cleaned up.
Suppose first that is the crystal of the defining representation of
sl n . This crystal is indexed by the set ng and e
defined if and
only if and in that case e
Next consider the tensor product (B (1)
)\Omega m . It may be indexed by words
in the alphabet [n], where u Its sl n crystal structure
is defined by the signature rule. This case of the signature rule is given in [18].
Now it is necessary to introduce notation for Young tableaux.
Some definitions are required. The Ferrers diagram D(-) is the set of pairs of
integers
g. A skew shape is the set
difference of the Ferrers diagrams D(-) and D(-) of the partitions - and -. If D
and E are skew shapes such that D has c columns and E has r rows, then define
the skew shape
In other words,
D\Omega E is the union of a translate of D located to the southwest of
a translate of E.
A tableau of (skew) shape D is a function
is depicted as a partial matrix whose (i; j)-th entry is T (i;
Denote by shape(T ) the domain of T . The tableau T is said to be column-strict
if T (i;
CST(D) be the set of column-strict tableaux of shape D.
The content of a tableau is the sequence
is the number of occurrences of the letter i in T . Let CST(D; fl)
denote the set of column-strict tableaux of shape D and content fl. The (row-
reading) word of the tableau T is the word given by word(T
is the word obtained by reading the i-th row of T from left to right. Say that the
word u fits the skew shape D if there is a column-strict tableau (necessarily unique)
whose row-reading word is u.
Remark 3. Let D be a skew shape, u a word in the alphabet [n] and 1
It is well-known and easy to verify that if ee i (u) is defined, then u fits D if and only
if ee i (u) does. This given, if T is a column-strict tableau of shape D and ee i (word(T
is defined, then let ee i (T ) be the unique column-strict tableau of shape D such that
)). The same can be done for e
Thus the set CST(D) is an sl n crystal; call it B D . When this is the
crystal B - .
3.4. Tensor products of irreducible sl n crystals and RSK. Let D be a skew
diagram and B D the sl n crystal defined in the previous section. The RSK correspondence
yields a combinatorial decomposition of B into irreducible sl n crystals.
The RSK map can be applied to tensor products of irreducible sl n crystals. The
goal of this section is to review a well-known parametrizing set for the multiplicity
space of such a tensor product, by what we shall call Littlewood-Richardson (LR)
tableaux.
be a sequence of positive integers summing to n and
sequence of partitions such that R i has j i parts, some of
which may be zero. Let A 1 be the first j 1 numbers in [n], A 2 the next j 2 , and so
on. Recall the skew shape
embedded in the plane so that A i gives
the set of row indices for R i . Let
is the length of
the i-th row of
\Delta\Omega R 1 , that is, fl is obtained by juxtaposing the partitions
. The tensor product crystal may be viewed as a skew crystal:
\Delta\Omega
for the weakly
increasing word (of length fl r ) comprising the r-th row of b, for 1 - r - n. The
word of b regarded as a skew column-strict tableau, is given by
Recall Knuth's equivalence on words [12]. Say that a skew shape is normal
(resp. antinormal) if it has a unique northwest (resp. southeast) corner cell [3]. A
normal skew shape is merely a translation in the plane of a partition shape, and
an antinormal shape is the 180-degree rotation of a normal shape. For any word
v, there is a unique (up to translation) column-strict tableau P (v) of normal shape
such that v is Knuth equivalent to the word of P (v). There is also a unique (up to
skew column-strict tableau P& (v) of antinormal shape such that v is
Knuth equivalent to the word of P& (v).
The tableau P (v) may be computed by Schensted's column-insertion algorithm
[27]. For a subinterval A ae [n] and a (skew) column-strict tableau T , define T j A to
be the skew column-strict tableau obtained by restricting T to A, that is, removing
from T the letters that are not in A. Define the pair of column-strict tableaux
By definition shape(Q(b)). It is easy to show that Q(b) is column-strict
and of content fl. This gives an embedding
CST(-) \Theta CST(-; fl)
(3.
It is well-known that this is a map of sl n crystals. That is, if g is any of e e i , e
For the case this fact is in [18].
Let us describe the image of the map (3.1). For the partition -
and a permutation w in the symmetric group Sn , the key tableau Key(w-) of
content w-, is the unique column-strict tableau of shape - and content w-. In the
above notation for the sequence of partitions R, for
in the alphabet A j .
Say that a word u in the alphabet [n] is R-LR (short for R-Littlewood-Richardson)
is the restriction of the word u to the
subalphabet A j ae [n]. Say that a (possibly skew) column-strict tableau is R-LR
if its row-reading word is. Denote by LRT(-; R) the R-LR tableaux of partition
shape - and
The following theorem is essentially a special case of [32, Theorem 1] which is a
strong version of the classical rule of Littlewood and Richardson [16].
Theorem 4. The map (3.1) gives a bijection
CST(-) \Theta LRT(-; R):
3.5. Kostka crystals. In the case that
, we call
R a Kostka crystal. The set LRT(-; R) is merely the set CST(-; fl).
be a sequence of rectangles as usual. Define the Kostka
crystal B rows(R) by the sequence of one row partitions rows(R) whose r-th partition
is given by (fl r ) where fl r is the length of the r-th row of the skew shape
Letting r the r-th row of b, there is the obvious sl n crystal
embedding
\Delta\Omega
In fact, the RSK correspondence (3.3) may be defined using the commutativity of
the diagram in (3.2) for
f i and all i 2 J , and giving its values on the sl n -highest
weight elements in B [23]. Suppose b 2 B is such that word(b) is of sl n -highest
weight, that is, ffl Such words are said to possess the lattice
property. In this case content(word(b)) must be a partition, say -, and
Key(-). For the recording tableau, write
r is the r-th row of b viewed as a tableau of the skew shape
Then Q(b) is the column-strict tableau of shape equal to content(b), whose i-th row
contains m copies of the letter j if and only if the word v j contains m copies of the
letter i, for all i and j. In particular, if elements the same sequences
of raising operators, then
3.6. Rectangle-switching bijections. From now on we consider only crystals
is the partition with j j rows and - j columns for 1 - j - m.
Consider the case
. Since the U q (sl n )-module V
R2\Omega
follows that there is a unique sl n crystal
isomorphism
It is defined explicitly in terms of the RSK correspondence as follows. By the above
multiplicity-freeness, for any partition -,
(R 1
Thus there is a unique bijection
For later use, extend - to a bijection from the set of (R 1 words to the set
of (R 2 words by
where Q(u) is the standard column-insertion recording tableau (that is,
Q(u) with u regarded as a word in the tensor product (B (1)
)\Omega N where N is the
length of u).
- is the rectangular generalization of an automorphism of conjugation. oe is
defined by the commutative diagram
R1\Omega
\Gamma\Gamma\Gamma\Gamma!
oe
R2\Omega
\Gamma\Gamma\Gamma\Gamma!
In other words, for all
Now the tensor product of rectangular
crystals. Let w 2 Sm be a permutation in the symmetric group on m letters. Write
where wR is the sequence of rectangles
Write oe R i ;R j for the action of the above sl n crystal isomorphism at consecutive
tensor positions in
\Delta\Omega
. Then the isomorphisms oe R i ;R j
satisfy a Yang-Baxter identity
j\Omega id)
This is a consequence of the corresponding difficult identity for bijections - R i ;R j
on recording tableaux, defined and conjectured in [13] and proven in [24, Theorem
9 (A5)]. By composing maps of the form oe R i ;R j , it is possible to well-define sl n
crystal isomorphisms
such that oe These bijections satisfy
P(oe R;wR
Q(oe R;wR
(3.
is the shape-preserving bijection defined in
[24].
Remark 5. Suppose is a Kostka crystal, b 2 B, and
with with R weakly increasing word of length fl j for 1
\Delta\Omega b 0
1 . In this case - R i+1 ;R i is
an automorphism of conjugation acting in multiplicity space. By definition b 0
It follows that b 0 can be computed from
b by a jeu-de-taquin on the two row skew tableau with word b i+1 b i .
3.7. b
sl n crystal structure on rectangular crystals. Suppose the sequence R
consists of a single rectangular partition with k rows and l columns, so that B
k;l . In [7] the existence of a unique classical b
sl n crystal structure on B k;l was
proved. The sl n crystal structure has already been described in Section 3.3. Using
the properties of the perfect crystal B k;l given in [7], an explicit tableau construction
for e
e 0 is presented.
The Dynkin diagram of b
sl n admits the rotation automorphism that sends i to
It follows that there is a bijection such that the
following diagram commutes for all i 2 I:
e
where subscripts are taken modulo n. Of course it is equivalent to require that /
satisfy the diagram with ee i replacing e
Lemma 6. / is unique and rotates content in the sense that for all
for all is equal to mn by convention.
Proof. Let b be the sl n -highest weight vector in B k;l , given explicitly by Key((l k )).
By the definition of / and the connectedness of B k;l it is enough to show that /(b)
is uniquely determined. By definition ffl i . Recall from [7] that
for all b
is said to be minimal if equality holds,
and that for any sequence of numbers (a
that sum to l, there is a
unique minimal vector b 0 such that ffl i (b
These facts imply that ffl 0 l. From the definition of / it follows that
Thus /(b) is minimal and hence uniquely defined.
Next it is shown that / is uniquely defined by a weaker condition than (3.9).
Lemma 7. / is uniquely defined by (3.9) for
Proof. Let the restriction of b to the subinterval [2;
[3;n] . By abuse of notation we shall occasionally identify a
(skew) tableau with its row-reading word.
If u is a word or tableau and p is an integer, denote by u+p the word or tableau
whose entries are obtained from those of u by adding p. The first goal is to show
that b b and b b are Knuth-equivalent, that is, P 1). By the assumption
on /, b b admits a sequence of lowering operators e
only if b b admits the sequence e i 1
are words in the
alphabet proves that P characterization
of the RSK map (see section 3.5).
Now the shape of the tableau b 0 is a rectangle, so its restriction b b
[3;n] to
a final subinterval of [n], has antinormal shape. Hence b b
uniquely determined by b b.
It only remains to show that the subtableau b 0 j [1;2] is uniquely specified. Its
shape must be the partition shape given by the complement in the rectangle (l k )
with the shape of b 0
[3;n] . Now b 0 j [1;2] is a column-strict tableau of partition shape and
contains only ones and twos, so it has at most two rows and is therefore uniquely
determined by its content. But its content is specified by (3.10).
The map / is explicitly constructed by exhibiting a map that satisfies the conditions
in Lemma 7.
The following operation is Sch-utzenberger's promotion operator, which was defined
on standard tableaux but has an obvious extension to column-strict tableaux
[3] [29]. Let D be a skew shape and b 2 CST(D). The promotion operator applied
to b is computed by the following algorithm.
1. Remove all the letters n in b, which removes from D a horizontal strip H
(skew shape such that each column contains at most one cell).
2. Slide (using Sch-utzenberger's jeu-de-taquin [3] [28]) the remaining subtableau
bj [n\Gamma1] to the southeast into the horizontal strip H, entering the cells of H
from left to right.
3. Fill in the vacated cells with zeros.
4. Add one to each entry.
The resulting tableau is denoted pr(b) 2 CST(D) and is called the promotion of
the tableau b.
Proposition 8. The map / of (3.9) is given by pr.
Proof. pr is content-rotating (satisfies (3.10)) and satisfies (3.9) for
since pr(t)j commutes with sl n crystal operators. By
Lemma
In light of (3.9), the operators ee 0 and e
f 0 on B k;l are given explicitly by
ee
e
Remark 9. Consider again the map pr on b 2 B k;l . The tableau b
partition shape - := (l Its row-reading word has
Schensted tableau pair P( b Key(-). Let
which has antinormal shape and whose complementary shape inside the rectangle
(l k ) must be a single row of length p, that is, shape( b b 0
and Q( b b 0 ) is a column-strict tableau of shape - and content (l \Gamma
is the longest element of the symmetric group S k . So
Let D be the skew shape
R be the tensor product of
rectangular crystals,
Note that the operator pr on
B may be described by
\Delta\Omega
By the definition of e e 0 on a rectangular crystal and the signature rule, it follows
that
ee
e
as operators on
Example 10. Let given by the
following skew tableau of shape R
\Theta \Theta \Theta \Theta \Theta \Theta 1 1
\Theta \Theta \Theta \Theta \Theta \Theta 2 2
\Theta \Theta \Theta 1 1 3
\Theta \Theta \Theta 2 3 4
\Theta \Theta \Theta 3 4 5
The element pr(b) is given by
\Theta \Theta \Theta \Theta \Theta \Theta 2 2
\Theta \Theta \Theta \Theta \Theta \Theta 3 3
\Theta \Theta \Theta 2 2 4
\Theta \Theta \Theta 3 4 5
\Theta \Theta \Theta 4 5 6
The signature for calculating ee 1 on pr(b) is
must be applied to the second tensor position. Then ee 1 (pr(b)) equals
\Theta \Theta \Theta \Theta \Theta \Theta 2 2
\Theta \Theta \Theta \Theta \Theta \Theta 3 3
\Theta \Theta \Theta 1 2 4
\Theta \Theta \Theta 3 4 5
\Theta \Theta \Theta 4 5 6
Finally ee 0
\Theta \Theta \Theta \Theta \Theta \Theta 1 1
\Theta \Theta \Theta \Theta \Theta \Theta 2 2
\Theta \Theta \Theta 1 3 3
\Theta \Theta \Theta 2 4 4
\Theta \Theta \Theta 3 5 7
3.8. Action of e e 0 on the tableau pair. In this section an algorithm is given
to compute the tableau pair (P(ee 0 (b)); Q(ee 0 (b))) of e e 0 (b) directly in terms of the
tableau pair (P(b); Q(b)) of b. In light of (3.12) and (3.2) with
to give P(pr(b)) and Q(pr(b)) in terms of P(b) and Q(b).
\Delta\Omega pr(b 1 ) can be constructed by applying Remark 9 to each
tensor factor. The element b is regarded as a skew tableau of shape
\Delta\Omega b b
denote the shape of b b j , so that b b has shape
\Delta\Omega D 1 .
Next, let b b 0
j be the tableau of skew shape D 0
obtained from b b j
as in Remark 9. Write b b
1 , a skew column-strict tableau of shape
\Delta\Omega D 0
1 . Finally, pr(b) is obtained by adjoining zeros to b b 0 at the
vacated positions of the shape
\Delta\Omega R 1 that are not in D 0 , and then adding
one to each entry.
In other words, P( b b) is obtained from P(b) by removing the letters n, which occupy
a horizontal strip (call it H). It is well-known that Q( b b) is obtained from Q(b) by
reverse column insertions at the cells of H starting with the rightmost cell of H
and proceeding to the left, ejecting a weakly increasing word v of length mn (b).
Another way to say this is that there is a unique weakly increasing word v of length
mn (b) such that P (vQ( b Q(b). So the content of v is the difference of the
contents of Q(b) and Q( b b). Since Q(b) 2 LRT(R), it follows that v is the weakly
increasing word comprised of mn (b j ) copies of the maximum letter of A j for all j.
In light of Remarks 5 and 9,
where w R
0 is the automorphism of conjugation corresponding to the longest element
in the Young subgroup SA1 \Theta \Delta \Delta \Delta \Theta SAm in the symmetric group Sn . Recall that
be the skew shape given by the difference of the shapes
of P(pr(b)) and P(pr(b)j [2;n] ). It is well-known that Q(pr(b)j [2;n] ) is obtained from
Q(pr(b)) by reverse row insertions at the cells of H 1 starting from the rightmost
and proceeding to the left. Let u be the ejected word. Then using an argument
similar to that above, P (Q(pr(b)j [2;n] and u is the weakly increasing
word comprised of mn (b j ) copies of the minimal letter of A j for all j. Since u and
are both weakly increasing words it is easy to calculate directly that
v.
14 MARK SHIMOZONO
Therefore
Remark 11. In summary, the tableau pair (P(pr(b)); Q(pr(b))) is constructed from
the tableau pair (P(b); Q(b)) by the following steps. Let
1. Let H be the horizontal strip given by the positions of the letters n in P .
2. Let v be the weakly increasing word and b
Q the tableau such that shape( b
H, such that
Q). These may be produced by reverse
column insertions on Q at H from right to left.
3. Then
4. Let H 1 be the horizontal strip
Q).
5. Let P 1 be the column-strict tableau given by adjoining to P j [n\Gamma1] the letters
n at the cells of H 1 .
By [24, Proposition 15],
(w Rb
R (v b
where -R is the LR analogue of the right circular shift of a word by positions defined
in [24].
All of these steps are invertible, so a description of pr \Gamma1 is obtained as well.
Example 12. Continuing the previous example, the image of b under the map
(3.3) is given by the tableau pair
Then H is the skew shape given by the single cell (7; 1),
and
consists of the single cell (1; 5) and
6
3.9. The R-cocyclage and e
e 0 . In [24] the R-cocyclage relation was defined on the
set LRT(R). In the Kostka case this is a weak version of the dual of the cyclage poset
of Lascoux and Sch-utzenberger [18]. It is shown that every covering R-cocyclage
relation, realized as recording tableaux, is induced by e
e 0 on some element of B R .
Theorem 13. Let ux be an R-LR word with x a letter. Then there is an element
R such that that the
cell is not in the n-th row. In particular, every
R-cocyclage covering relation is realized by the action of e e 0 in this way.
Proof. By definition (see [24]), every covering relation in the R-cocyclage has the
form that P (v) covers P (-R (v)) where v is an R-LR word. It follows from [24,
Proposition 23] that if P is an R-cocyclage then x ? 1. If s is in
the n-th row, then by the column-strictness of P (ux) and the fact that all letters
are in the set [n], x = 1. So no R-cocyclage covering relations are excluded by the
restriction that s not be in the n-th row.
and the automorphism of conjugation w R
preserves shape, without loss
of generality it may be assumed that u is the row-reading word of a column-strict
tableau U of partition shape b shape(P(ux)). Now a skew
tableau t has partition shape if and only if is the word of the
column-strict tableau U
exists since Q 2 LRT(-; R)
and (3.3) is a bijection. It must be shown that Q(ee 0 This shall be
verified by applying the formula (3.12) and Remark 11.
Now n. The horizontal strip
H given by the cells of Key(w-) containing the letter n, is merely the n-th row
of the shape -. Since Q 2 CST(-) (and all tableaux are in the alphabet [n]), the
subtableau given by the first -n columns of Q is equal to Key((- n
the rest of Q and R 0 the sequence of rectangles obtained by removing the first
-n columns from each of the rectangles in R. Since Q is R-LR and the column-
reading word of Q equals that of Key((- n
y be the minimal element of the last interval Am . In the notation of Remark 11,
since Q r is R 0 -LR, it follows that
The right hand side is a column-strict tableau of shape
so that the horizontal strip H entirely in the first row,
and P(pr(b)) is formed from 1 by pushing the first row
to the right by -n cells and placing 1's in the vacated positions. The tableau
ones. Hence P(pr(b)) contains - t twos in its
first row, that is, ffl 1
which changes the letter 2 at the cell (1; mn (b) + 1) to a
1. By (3.2),
Now pr \Gamma1 is applied, reversing the algorithm in Remark 11 starting with the tableau
1 and H 0 for the analogues of H 1 and
H. By Remark 11 and direct calculation,
In particular H 0
[fsg. The reverse row insertions on Q(ee 1
at
merely remove the -n copies of y from the first row by (3.14). The final
reverse row insertion (at the cell s) stays within the subtableau Q r and changes it
to the tableau Q r " say, and ejects the letter x, since since x is obtained from Q
by reverse row insertion at s and
. The result of the reverse row
insertion on Q(ee 1 (pr(b))) at H 0
1 is
with ejected word xy -n . Writing U
r and using the fact that Q r is
Q(ee
Remark 14. Suppose B R is a Kostka crystal with R
n. Then Theorem 13 shows that every covering relation in the cyclage poset
- CST(-; fl) is realized in the recording tableaux by e e 0 acting on some b 2 B R .
Remark 15. Let the maximum number of columns among the
rectangles R i in R. Suppose is an sl n -highest weight vector such that Q(b)
has shape -, such that covering relation in
the R-cocyclage.
To see this, let us adopt the notation of the proof of Theorem 13. Let
be the corner cell in the last column of -. Then - so that w-. By the
choice of b,
To apply Theorem 13 it must be shown that t ! n. Suppose not. Then - n
and Q(b) is an R-LR tableau of shape (- n
the total of the heights of the
rectangles in R is n, it follows that all of the rectangles in R must have exactly - 1
columns, contradicting the assumption that - 1 ? M .
Apply the reverse row insertion on Q(b) at the cell s, obtaining the column-strict
tableau U of shape - \Gamma fsg and ejecting the letter x. Then P
is an R-cocyclage, by [24, Remark 17].
Example 16. Continuing the example, e e 0 (b) is computed below. Q(ee 1
Q(pr(b)) and
6
Applying pr \Gamma1 to the tableau pair of ee 1 (pr(b)) and denoting by P 0
the
corresponding tableaux and word, we obtain
and finally
P(ee
Remark 17. Suppose that is such that each R j has a common
number of columns, say l. Then B R , being the tensor product of perfect crystals
of level l, is perfect of level l and therefore connected by [6]. In this case, more
is true. Using Remark 15, every element can be connected to the unique sl n -highest
weight vector in B R of charge zero, by applying last column R-cocyclages on the
recording tableau.
For R a general sequence of rectangles, B R is still connected. However it is not
necessarily possible to use e
e 0 to connect every sl n -component to the zero energy
component in such a way that the energy always drops by one. For example,
and the sl 3 -component with Q-tableau 1 1
. The
applications of e e 0 on the five elements of this component that admit e e 0 , all produce
elements in the component with Q-tableau 1 1 3
2 which has the same energy.
3.10. Rectangle-switching bijections and ee 0 .
Proposition 18. Let R 1 and R 2 be rectangles. Then the rectangle-switching bijec-
tion
is an isomorphism of classical b
sl n crystals.
Proof. Since is known to be an isomorphism of sl n -crystals, it only
remains to show that oe commutes with ee 0 . Let b 2 B
. By the bijectivity
of (3.3) it is enough to show that
P(ee
Q(ee
Consider first the process in passing from (P(b); Q(b)) to (P(ee 0 (b)); Q(ee 0 (b))). Let
H and H 1 be the be the horizontal strips, v the weakly increasing word and b
the tableau as in Remark 11. Let b
the analogous
objects in passing from (P(b 0
shape(P(b)). Observe that P(b 0 by (3.6), so that
This implies that the increasing words v 0 and v
have the same length mn (b) and shape( b
call their common shape b -.
Now
is the unique element in LRT(-; (R 2
is the unique element in LRT(-; (R 1
On the other hand, consider the word v b
Q, identifying b
Q with its row-reading word.
The tableau Q(v b
Q) has shape -. Let p and q be the number of cells in b
respectively. Since b
Q is a column-strict tableau of shape b -, Q(v b
Q)j [p] is the rowwise
standard tableau of shape b -, the unique standard tableau of shape b - in which
is located immediately to the right of i not in the first column.
Now Q(v b
Q)jH is filled from left to right by the numbers since it
records the column-insertion of the weakly increasing word v. The same argument
applies to Q(v 0 b
that - (v b
0 and -R 0 for the corresponding
constructions for R 0 . By [24, Theorem 16],
(w R 0b
R (v b
Applying the P tableau part of (3.5) to the word (w Rb
By this and the Q tableau part of (3.6) for the word e e 0 (b),
Q(oe(ee
This proves the equality of the Q-tableaux in (3.16).
For the P-tableaux, let us recall the process that leads from (P(b); Q(b)) to
(P(ee 0 (b)); Q(ee 0 (b))) and from (P(b 0
Recall that P(b 0 they have the same restriction to the alphabet
On the other hand, since it has been shown that the tableaux Q(ee 0 (b))
and Q(ee 0 (b 0 )) have the same shape, it follows that the horizontal strips H 1 and H 0coincide. So P(ee 0 (b)) and P(ee 0 (b 0 restricted to the alphabet [2; n].
Since both tableaux also have the same partition shape they must coincide. This,
together with the P-tableau part of (3.6) applied to e e 0 (b), shows that
P(ee
3.11. Energy function. In this section it is shown that the energy function of the
classical b
sl n crystal B R is given by the generalized charge of [24] on the Q-tableau.
The definition of energy function follows [7] and [22].
Consider the unique classical b
sl n crystal isomorphism
An energy function
!Zis a function that satisfies the following
axioms:
such that e
and similarly for ee i .
(H2) For all
such that e
e 0 (b) is defined,
H(ee
defined in the same way then
Proof. Without loss of generality assume that - 1 - 2 . Let
be the partition whose Ferrers diagram is obtained by placing the shape R 1 atop R 2 .
Let QR be the unique element of the singleton set LRT(fl(R); (R 1
that fl(R) is the only shape admitting an R-LR tableau that has at most - 1 columns.
Let R be the unique sl n -highest weight vector of weight fl(R); it is given
explicitly by v
in the notation of the definition of R-LR, and satisfies
It is shown that every element b 2 B R is connected to v R . First, using sl n -raising
operators, it may be assumed that b is an sl n -highest weight vector. If Q(b) has
at most - 1 columns then Otherwise Q(b) has more than - 1 columns, and
Remark 15 applies. But e
e 0 (b) is closer to v R in the sense that Q(ee 0 (b)) has one
fewer cells to the right of the - 1 -th column than Q(b) does [24, Proposition 38], so
induction finishes the proof.
By Lemma 19 H is uniquely determined up to a global additive constant. H is
normalized by the condition
with v R as in Lemma 19. Equivalently,
where Key(R i ) is the highest-weight vector in B R i and the value of H is the size
of the rectangle R 1 " R 2 .
For a tableau Q 2 LRT(-; (R 1 to be the number of cells
in the shape of Q that are strictly east of the max(-
Proposition 20. Let be a pair of rectangles. Then for all
Proof. Follows immediately from the proof of Lemma 19.
Now consider
The energy function for B is given as follows. Denote by H Zgiven
by the value of the energy function H (wB) i+1 ;(wB) i at the (i + 1)-st and i-th tensor
positions (according to our convention). Recall the isomorphisms of classical b
crystals (3.7). Define the cyclic permutation w
1!j-m
Call the inner sum E (j)
R (b).
The following version of [9, Lemma 5.1] holds for ER with no additional difficulty.
Lemma 21. Let
\Delta\Omega b
Example 22. Let b be as in the running example. Then
\Theta \Theta \Theta \Theta \Theta \Theta 1 1
\Theta \Theta \Theta \Theta \Theta \Theta 2 2
\Theta \Theta \Theta 1 3 3
\Theta \Theta \Theta 2 4 4
so that
so that d 2 2. Finally
so that d 1 (- 2
3.12. Energy and generalized charge. Define the map ER
ER (b) for any b 2 B R such that This map is well-defined
since ER is constant on sl n -components and the map (3.3) is a bijection. It follows
immediately from the definitions that
1-i!j-m
The Kostka case of the following result was first proven by K. Kilpatrick and
D. White. In the further special case that - is a partition it was shown in [22]
that ER (Q) is the charge. Now in the Kostka case the generalized charge statistic
charge R specializes to the formula of charge in [15].
Theorem 23. charge
Proof. Let Q 2 LRT(R) of shape -, say. It will be shown by induction on the
number of rectangles m and then on charge R (Q), that that ER : LRT(R) ! Z
satisfies the intrinsic characterization of charge R by the properties (C1) through
(C4) [24, Theorem 21]. Let
First, (C2) need only be checked when Q last column R-cocyclage,
and (C4) need only be verified when - To see this, if then there is a
last column R-cocyclage Q 0 !R Q and in this case charge R (Q
. If (C3) does not apply, then one may apply (C4) several times
to switch the widest rectangle closer to the beginning of the sequence R and then
apply (C3), which decreases the number of rectangles m.
(C1) is trivial. For (C2), let Q 0 !R Q be a last-column R-cocyclage with
b such that
and as in Theorem 13 and Remark 15. By the proof of Theorem 13 in
this case, ffl 0
However, Q(ee 0 and (C2) has been
verified.
To check (C3), let b
of
R). It follows that
22 MARK SHIMOZONO
for all j ? 1. Therefore
1-i!j-m
which verifies (C3).
For (C4), the proof may be reduce to the case 3. By abuse of notation we
suppress the notation for the sequence of rectangles, writing - p (Q) for the operator
that acts on the restriction of an LR tableau to the p-th and (p 1)-st alphabets,
and similarly for the function d p . Write w i;j := -
d i;j := d i (w i;j Q):
The value d 0
i;j is computed using a case by case analysis.
1. In this case it is clear that d 0
2.
3.
so that
4. is the identity, and
5. so that
since the restriction of w i;j Q to the i-th and i 1-st subalphabets is not
affected by - p+1 .
7.
8. p. In this case it is clear that d
i;j .
Based on these computations, the difference in energies E -p R (-
as follows. In cases 1, 4, and 5, and 8, d 0
so these terms cancel. The sum
of the terms in cases 6 and 7 cancel. So it is enough to show that the sum of terms
in 2 and 3 cancel, that is,
Rewriting d p (w p;j observe that without loss of generality it
may be assumed that must be shown that
Recall that in verifying (C4) it may be assumed that - . In this case [24,
Remark 39] applies. Say - There
are three cases, namely four terms in
are zero. If then the first and fourth terms are zero and the second
and third agree, while if then the second and third are zero and the first and
fourth agree.
Corollary 24.
chV wt sl (-) K -;R (q);
where - runs over partitions of length at most n.
Proof. The equality follows immediately from Theorem 23, the weight-preserving
bijection (3.3), and [24, Theorem 11].
4. Tensor product structure on Demazure crystals
The tensor product structure for the Demazure crystals, is a consequence of an
inhomogeneous version of [10, Theorem 2.3] that uses Lemma 21.
Theorem 25. Let - be a partition of n. Then
Bw- (l  0
;l\Omega u l  0
as classical b
sl n crystals, where the affine b
sl n Demazure crystal is viewed as a classical
sl n -crystal by composing its weight function wt with the projection cl .
Moreover, if v 7!
b\Omega u l  0
then
where the left hand side is the distance along the null root ffi of v from the highest
weight vector u l  0 2 V(l  0 ) and R is defined by R
5. Proof of Theorem 2
Theorem 2 follows from Theorem 25 and Corollary 24.
6. Generalization of Han's monotonicity for Kostka-Foulkes
polynomials
The following monotonicity property for the Kostka-Foulkes polynomials was
proved by G.-N. Han [4]:
denotes the partition obtained by adding a row of length a to -.
Here is the generalization of this result for the polynomials K -;R (q) that was
conjectured by A. N. Kirillov.
Theorem 26. Let R be a dominant sequence of rectangles and
tangle. Then
is the partition obtained by adding m rows of size k to - and R[(k m )
is any dominant sequence of rectangles obtained by adding the rectangle (k m ) to R.
Proof. Write
the map
the letters of Y 0 are smaller than those of Q
follows that shape(i R since it is Knuth
equivalent to a shuffle of Y 0 and the tableau Q+m, which is R-LR in the alphabet
+m]. Thus the map i R is well-defined. Let B represent the union of the
zero-th and first subalphabets for w Y be the key tableau for the first
subalphabet of w
by the Knuth invariance of d 0 , the fact that w 0;j doesn't touch letters in the zero-th
subalphabet, the definition of d 0 , the fact that w 0;j Q 2 LRT(w 0;j R), and direct
calculation of the shape of P (Y Y 0 ) combined with Proposition 20. If i ? 0 then
From (6.1) and (6.2) it follows that E R +(i R



--R

Combinatorial properties of partially ordered sets associated with partitions and finite Abelian groups

Dual equivalence with applications
Croissance des polyn-omes de Kostka
Infinte dimensional Lie algebras
Modern Phys.
Perfect crystals of quantum affine Lie algebras
On the Grothendieck group of modules supported in a nilpotent orbit in the Lie algebra gl(n)


Crystal graphs for representations of the q-analogue of classical Lie algebras

A generalization of the Kostka-Foulkes polynomials
Cyclic permutations on words
Crystal graphs and q-analogues of weight multiplicities for root systems of type An
Group characters and algebra
Sur une conjecture de H.
in Noncommutative structures in algebra and geometric combinatorics
Green Polynomials and Singularities of Unipotent Classes

The characters of the group GL(n


A cyclage poset structure for Littlewood-Richardson tableaux

generalized Kostka-Foulkes poly- nomials
Math 13
correspondance de Robinson
Promotion des morphisms d'ensembles ordonnes
Bases of coordinate rings of closures of conjugacy classes of nilpotent matrix
Characters of modules supported in the conjugacy class of a nilpotent matrix
The equations of conjugacy classes of nilpotent matrices
Some connections between the Littlewood-Richardson rule and the construction of Schensted
--TR
Dual equivalence with applications, including a conjecture of Proctor
Graded characters of modules supported in the closure of a nilpotent conjugacy class
A cyclage poset structure for LittlewoodMYAMPERSANDmdash;Richardson tableaux
Multi-atoms and monotonicity of generalized Kostka polynomials

--CTR
Lipika Deka , Anne Schilling, New fermionic formula for unrestricted Kostka polynomials, Journal of Combinatorial Theory Series A, v.113 n.7, p.1435-1461, October 2006
Anne Schilling , Philip Sternberg, Finite-dimensional crystals B2,s for quantum affine algebras of type D(1)n, Journal of Algebraic Combinatorics: An International Journal, v.23 n.4, p.317-354, June      2006
S. Ole Warnaar, The Bailey Lemma and Kostka Polynomials, Journal of Algebraic Combinatorics: An International Journal, v.20 n.2, p.131-171, September 2004
tableau;littlewood-richardson coefficient;crystal graph;kostka polynomial