External Littelmann paths for crystals of type A
Ola Amara-Omari Mary Schaps
math.RT 17B10, 17B65



EXTERNAL LITTELMANN PATHS OF TYPE A]EXTERNAL LITTELMANN PATHS FOR CRYSTALS  OF TYPE A

 Bar-Ilan University, Ramat-Gan, Israel,
mschaps@macs.biu.ac.il

 Bar-Ilan University, Ramat-Gan, Israel,
 olaomari77@hotmail.com
 [2010] 17B10, 17B35
^1Partially supported by Ministry of Science, Technology and Space fellowship , at Bar-Ilan University.

[
    Ola Amara-Omari^1 and  Mary Schaps
    Received: date / Accepted: date
======================================










  
For the Kashiwara crystal of a highest weight representation of an affine Lie algebra of  type A and rank e, with highest weight Λ, there is a labeling by multipartitions and by piecewise linear paths in the real weight space called Littelmann paths.  Both labelings are constructed recursively, but since Kashiwara demonstrated that the crystals are isomorphic, there is a bijection between the labels.  

We choose a multicharge (k_1,…,k_r), with 0 ≤ k_1≤  k_2....≤ k_r ≤ e-1. We put k_i in the node at the upper left corner of partition i of the multipartition and let the residues from ℤ/ e ℤ increase across rows and decrease down columns.    For e=2, we call a multipartition residue-homogeneous if all nonzero rows end in nodes of the same residue and partitions with the same corner residue have first rows of the same parity. It is strongly residue homogeneous if each partition ends in a triangle of whose side has length one less than the first row of the next partition.

In this paper we show that each such multipartition corresponds to a Littelmann path which is unidirectional in the sense that the projection of the the main part of the path to the coordinates of the fundamental weights consists of long paths all lying in either the second or fourth quadrant, separated by oscillating paths with a fixed integer oscillator. The path corresponding to such a multipartition can be constructed non-recursively using only integers describing the structure of the multipartition.





§ INTRODUCTION


Let e be the rank of an affine Lie algebra  𝔤 of type A and let r be the level of a highest weight module with highest weight Λ. For type A, the level r  is the sum of the coefficients of the fundamental weights in the dominant integral weight  Λ. There is a labelling of the elements of the basis of the highest weight module by multipartitions and those multipartitions which are recursively constructed as labels of the basis elements are called e-regular.  The level r is the common number of partitions in these multipartitions.

 The problem of determining the e-regular multipartions of type A in a non-recursive manner has shown only slow progress. It is known for r=1.  Mathas settled it for e=2 in <cit.>, and Ariki, Kreiman and Tsuchioka for r=2 in <cit.>.  A few other results extending these are also available.  We are trying to attack the problem through Littelmann paths, by trying to find a direct way to translate between the multipartition labelling  a crystal basis element and the corresponding Littelmann path. The results in this paper are for the e=2 case, for which, as we mentioned, Mathas already solved the problem of finding a criterion for e-regular multipartitions, but we hope that finding a way to translate from Littelmann paths to multipartitions will be relevant to the problem of e-regularity, since only e-regular multipartitions will correspond to a Littelmann path.

In 2 and 3, we give needed background, and the definition of a unidirectional Littelmann path.  In 4, we define a strongly residue homogeneous multipartition and prove our main result, that a strongly residue-homogeneous multipartition corresponds to a unidirectional Littelman path whose invariants can be determined from the structure of the multipartition. In 5, we give examples to illustrate the limitations of the theorem and explain what remains open.



§ DEFINITIONS AND NOTATION


Let 𝔤 be the affine Lie algebra A^(1)_e-1. Let C be the Cartan matrix, and δ the null root. Let Λ be a dominant integral weight, let V(Λ) be the highest weight module with that 
highest weight, and let P(Λ) be the set of weights of V(Λ). Let I be the set of residues ℤ / e ℤ and let  Q be the ℤ-lattice generated by the simple roots,

    _0,…,_e-1.

Let  Q_+ be the subset of Q in which all coefficients are non-negative. 

The weight space P of the affine Lie algebra has two different bases.  One is given by the fundamental weights together with the null root, Λ_0,…, Λ_e-1, δ, 
and one is given by Λ_0, _0,…,_e-1.  We will usually use the first basis for our weights.  

A highest weight module V(Λ) is integrable.  Every weight λ has the form Λ-α, for α∈ Q_+.  The vector
of nonnegative integers giving the coefficients of α is called the content of λ. We will not repeat all the standard material about the symmetric form (- | -), which can be found in <cit.>. We follow <cit.> in defining the defect of a weight by


    (λ)=1/2((Λ|Λ)-  (λ|λ)).



 Since we are in a highest weight module, we always have (Λ|Λ) ≥ (λ|λ), and the defect is in fact an integer for the affine Lie algebras of type A treated in this paper. The weights of defect 0 are those lying in the Weyl group orbit of Λ and will play an important role in the definition of the Littelmann paths. 
Define

    max P(Λ)={λ∈ P(Λ) |λ + δ∉P(Λ)},
 
and by <cit.>, every element of 
P(Λ) is of the form {y-kδ| y ∈max P(Λ), k ∈ℤ_≥ 0}.
Let W denote the Weyl group, generated by reflections s_0,…, s_e-1.

By the ground-breaking work of Chuang and Rouquier <cit.>, the highest weight module V(Λ) can be categorified.  The weight spaces lift to categories of representations of  blocks of cyclotomic Hecke algebras, the basis vectors lift to simple modules,  the Chevalley generators e_i,f_i lift to restriction and induction functors E_i,F_i, and the simple reflections in the Weyl group lift to derived equivalences, which in a few important cases are actually Morita equivalences.  We will not be using the categorified version, but it provides the underlying motivation for trying to understand the multipartitions, which label the simple modules of the cyclotomic Hecke algebras. For r=1, in what is called the degenerate case, these are the group algebras of symmetric groups.

There are three distinct labellings for an element of a Kashiwara crystal B(Λ), <cit.>,<cit.>, of type A.  We will not directly need the definition of the crystal or of its crystal base. An element of the crystal base can be labelled  by its Littelmann path,  by its multipartitions, and by its canonical basis.  In this paper we are concerned only with the first two, looking for cases where there is a direct connection between them, so that we can read off the multipartition from the Littelmann path or the Littelmann path from the  multipartition.  In a different paper, <cit.>, we make a similar direct passage from a multipartition to a canonical basis element for a symmetric crystal.






The theory originated with work of Lakshmibai and Seshadri for type A, but it was Littelmann in <cit.> who extended to other types and  who proved many of the most important properties of the path model of the highest weight representations. 
A path π is a continuous, piecewise linear function from the closed real interval [0,1] into the real space ℝ⊗_ℤ P(Λ) such that π(0)={0} and π(1) ∈ P(Λ).  The weight π(1)   will be called the weight of the path.


 For any residue ϵ in the set of residues I, we define a function H_ϵ^π(t)=π(t),h_ϵ, which is simply the projection of the path onto the coefficient of Λ_ϵ.  We then set

    m_ϵ=min_t (H^π_ϵ(t)).


This minimum is always achieved at one of the finite set of corner weights and is always non-positive, since  H_ϵ^π(0)=0.  We let 𝒫_int be the set of paths for which this 
m_ϵ is an integer for all ϵ∈ I. Littelmann proves in <cit.>, Lemma 4.5(d), that all the Littelmann paths in the crystal for a dominant weight Λ lie in P_int. Note that since π(1) ∈ P(Λ), all the numbers H_ϵ^π(1) are integers. 

 Littelmann's function f_ϵ is given on 𝒫_int as follows:


  * If H^π_ϵ(1)=m_ϵ, then f_ϵ(π)=0.

  * Set

    t_0=   max_t { t ∈ [0,1] | H^π_ϵ (t)=m_ϵ}
    
    t_1=   min_t { t ∈ [t_0,1] | H^π_ϵ (t)=m_ϵ+1  }

then if H^π_ϵ(t) is monotonically increasing on the interval [t_0,t_1], we define

    f_ϵ(π)(t)=
    π(t)    t ∈ [0,t_0]
    π(t_0)+s_ϵ(π(t)-π(t_0))    t ∈ [t_0,t_1]
    π(t)-α_ϵ   t ∈ [t_1,1]

In the more complicated case, where the path  H^π_ϵ(t) is not monotonic  from t_0 to t_1, but contains some segments which oscillate, can be found in <cit.>. We will have oscillating sections of the path, but they will all occur before t_0 or after t_1.


The definition of e_ϵ is dual and reverses the action of f_ϵ. The definition is given in <cit.>. We will be using primarily f_ϵ.

 Let 𝒫 denote the set of paths. For ν,μ defect 0 weights, we define ν≥μ if there is a sequence of defect 0 weights 
ν=ν_0,...,ν_s=μ and positive real roots β_1,…,β_s such that  

    ν_i =s_β_i(v_i-1)

and
 
    ν_i-1,β_i^∨ <0, i=1,2,…,s.


Following a suggestion of Kashiwara, Littelmann then defines a distance function between ν and μ.  We do not need the entire function and its properties, only the case dist(μ,ν)=1, which means that no other weight can be inserted between them preserving the order.

 <cit.> An a-chain for (μ,ν) is a sequence μ=λ_0>λ_1>…>λ_s=ν of defect 0 weights such that either s=0 and μ=ν or λ_i=s_β_i(λ_i-1) for some positive real roots  β_1,…,β_s  with dist(λ_i,λ_i-1)=1 and 
aλ_i ,h_β_i∈ℤ




 <cit.> A  rational path  of class Λ is an ordered set of sequences (ν; h)   for which  following hold:


  * ν_1>…>ν_s is a linearly ordered set of defect 0 weights in WΛ. 

  * 0<b_1<b_2<…<b_s=1 are rational numbers.  We will denote 0 by b_0.

We identify π with the parameterized path 

    π(t)=∑^j-1_ℓ=1 (b_ℓ-b_ℓ -1)ν_ℓ +(t-b_j-1)ν_j,  b_j-1≤ t ≤ b_j


  A rational path (ν, b) is called an LS-path if there is an b_i-chain connecting ν_i to ν_i+1.
We will generally write the rational path in the form 


    (ν_1,…, ν_s ;b_1,b_2,…,b_s=1).

and call it  an LS-representation of the path. In this paper, the numbers b_i will be endpoints of subintervals of [0,1] parameterizing straight subpaths and will be called parameter endpoints.
 In Example <ref>, we give an example of a rational path with non-trivial b_i-chains.  In <cit.> the authors refer to ν_1 as the ceiling and ν_s as the  floor.  Since there is a unique multipartition for a defect 0 weight, they sometimes use the same notation for the corresponding multipartitions but in this article the ceiling and floor will always be weights.


 Let B=B(Λ) be the crystal of a dominant integral weight Λ. For any v  ∈ B, we let θ(v)=(θ_0,…, θ_e-1) be the hub of v, where 

    θ_i= wt(v),α_i^∨

The hub is the projection of the weight of v onto the subspace of the weight space generated by the fundamental weights <cit.>.
 


By doing numerous calculations of Littelmann paths, we found one particular class, which we call unidirectional,  that was particularly easy to handle. 
We consider the monotone sequence of weights of our rank 2 affine Lie algebra, ψ_1=s_0Λ,   ψ_2=s_1s_0Λ, …ψ_m=… s_0s_1s_0Λ, where m is the number of reflections. For each m we define an integer d_m=rm-b, which will be the number of rows in the multipartion corresponding to that weight. We will demonstrate later that the hub θ_m of ψ_m is [-d_m, d_m+1] if m is odd and [d_m+1,-d_m] if m is even.

 A Littelmann path for type A and rank 2 will be called unidirectional  if there is an integer k ≥ 0 and a strictly descending sequence of positive integers of the same parity, m_1,m_2,… m_k,  a sequence of positive integers, I_m_1, I_m_2,… I_m_k, an integer m_k+1 with 0 ≤ m_k+1<m_k,  and a non-negative sequence of integers C_1,C_2,…, C_k+1, such that  the projection of the path onto the hubs consists of two parts:


  * The main part, which is entirely contained in the second or fourth quadrant.  Let ϵ be the residue corresponding to the positive coordinate of the quadrant.  Its hubs are θ_m for m_k+1≤ m ≤ m_1. so that ϵ =1 in the second quadrant and ϵ=0 in the fourth quadrant. Starting at the origin, the main part consists of k long paths of  positive integral ϵ-length I_m_i, which are multiples of θ_m_i, i ≤ k, followed by a sequence of paths of the form C_i/d_md_m+1θ_m, for m=m_i,m_i-1,…,m_i+1+1, which we call oscillating paths. Note that, if C_i ≠ 0 so that the set of oscillating paths is non-empty, the first of the oscillating paths is a continuation of the long path with index i.

  * The seed part, whose hub may contain or not contain any of the following:

 
  * A transit path  1/d_m_k+1+1θ_m_k+1, which may be embedded in a longer path.

  * A tail, which is a multiple of θ_0 or of θ_1

  * A long path which is a multiple of θ_1 or θ_2, , together with an oscillating path.

  * An oscillating set of straight paths, multiples of θ_u for u ≤ m_k+1 with coefficient C_k+1/d_ud_u+1, where C_k=C_k+1+d_m_k+1.                                                              

In our main theorem, we will determine the exact significance of these elements of the seed part, but the intention of the name is that the transit path and the tail are the seeds from which new long paths  can grow.





§ MULTIPARTITIONS


We now turn to the second labelling which will concern us in this paper, the multipartitions.  Unlike the Littelmann paths, this labelling is available only is the important case of type A.  Also unlike the Littelmann paths, it requires some non-canonical choices, a choice between two dual versions of the theory and the choice  of an object called the multicharge. Once these choices have been made, the aim of the paper will be to find conditions on the multipartition which insure that the corresponding Littelmann path will be unidirectional. We will then be able to read off the structure of the Littelmann path directly from the multipartition.





A multicharge will be a sequence s=(k_1,k_2,…,k_r) of integers with 0 ≤ k_i ≤ e-1, for some natural number r which will be called the level of the multicharge. 

A multicharge s determines a dominant integral weight

    Λ=Λ_k_1+…+Λ_k_r.

In the abelian weight space the order of the summands is of course irrelevant; it is simply a notational reminder of the multicharge. We will follow Mathas in <cit.> in requiring k_1 ≤ k_2 ≤…≤ k_r.  We can then summarize by setting 

    Λ=a_0 Λ_0+…+a_e-1Λ_e-1


 We can regard the integers k_i as  elements of ℤ / e ℤ, and with a slight abuse of notation we will identify k_i with this residue. 
We have to assume that we know the ceiling and floor of an multipartition, constructed  in <cit.> from an LS-representation by letting the ceiling be ν_1 and the floor be ν_s. 


 The Young diagram of the multipartition of a defect 0 weight λ will be represented by Y(λ). If the t-th partition λ^t  of λ is nonempty, then we associate to each node in the Young diagram a residue, where the node (i,j) is given residue

    k_t +j-i

This will be called a k_t-corner partition.  Due to the Mathas condition on the ordering of the k_i, this means that the multipartition will consist of a_o 0-corner partitions, followed by a_1 1-corner partitions and so forth.




 §.§ Standard LS-paths


The main theorem of this paper will be a proof that a strongly residue homogeneous multipartition corresponds to a unidirectional Littelmann path. However, we do more:  we show that the LS-repesentation of the Littelmann path can be written down  directly from the Young diagram of the multipartition. This close connection between the multipartiton and the Littelmann path will be crucial in carrying out the induction in  our main theorem. An LS-representation for which both the defect 0 weights and the parameter endpoints can be determined from the multipartition in the way we describe  will be called standard.   We will give the definition in general for any rank, but will apply it only in the case e=2.   We must first prepare considerable notation.

 An important part of the description of each partition λ^t in the multipartition λ is the length a(λ^t) of the first row. Since these lengths decrease from the ceiling to the floor, the ascending numbering from ceiling to floor used in 
  <cit.> or <cit.> is not compatible with our notation, so we will reverse the numbering.  We  assume that  we have fixed a  reduced word s_i_q… s_i_1 and the sequence of defect 0 weights μ=μ_q, μ_q-1,…, μ_0, with μ_j=s_i_jμ_j-1,  going back to Λ. By the partial ordering of defect 0 weights which we introduced above, we thus have μ_j > μ_j-1 and in fact, since we are using simple roots, they  will have dist(μ_j , μ_j-1)=1. That will be important, because our a-chains will be strings of these μ_j.


 If we let d_j be the number of times we must operate on μ_j-1 with f_i_j to produce the reflection s_i_j, then in terms of the multipartitions, we have 

    d_j=#(Y(μ_j)-Y(μ_j-1))

that is to say, the number of nodes in the difference between the two Young diagrams. The d_j increase as j increases.  Since all the added nodes have the same residue, we never add more than one to any given row. Now we consider the set

    ((Y(μ_j)-Y(μ_j-1))∩ Y(λ))

which is called a ladder in the article by Fayers <cit.> generalizing the LLT-algorithm <cit.> for canonical bases. If the ladder intersects the top row of the multipartition at node m of residue i, we will call this the m-ladder, whereas for Fayers it was an i-ladder.  Not only the proof but even the statement of our main theorem will depend on the lengths of these ladders, which we denote by

    c_m=#(((Y(μ_j)-Y(μ_j-1))∩ Y(λ)).

Clearly c_j ≤ d_j for all j.  However, the c_i do not necessarily increase as j increases.


 Given a choice of reduced word and, as above,  a sequence  μ_q, μ_q-1,…, μ_0 of defect 0 weights  set

    e_j=c_j/d_j

for j>0 and set   e_0=1. Let 

    j_p, j_p-1,…,j_0
 
 be the subsequence of q, q-1,…,0 of j for which  e_j ≠ e_j+1
 so the μ_j_0 will be the floor.  Then the LS-representation

    (μ_j_p,…, μ_j_0;e_j_p,…,e_j_0)

will be called standard with respect to the multipartition λ if the sequence of e_i is increasing and the Littelmann path determined by this LS-representation corresponds in the crystal to the multipartition.


(A standard LS-representation with gaps)
Let e=3. let Λ=3Λ_0  and set λ =[(8,6,1),(),()]. From our computer calculations, we know that the ceiling is the defect zero  multipartition with  weight 
-12Λ_1 +15Λ_2-24 δ. This  corresponds to three copies of the partition (8,6,4,2), and equals three times the weight of a single copy of that partition in B(Λ_0). This ceiling has a periodic representation as s_1s_0s_2s_1s_0s_2s_1s_0Λ. The first ladder is the 8-ladder with residue 1, consisting of positions 8 and 6 in the first two rows.  For m=7, we also get an m-ladder of length 2, consisting of positions 7 and 5 in the first two rows, and similarly for m=6,  we get a ladder of length 2, so that c_8=c_7=c_6=2. Now, however, with the 5-ladder of residue 1 we get c_5=3. 

 If we calculate the d_j, descending from 8 to 1, we get 12, 12,9,9,6,6,3,3.  The 
c_j in the same order are 2,2,2,3,2,2,1,1. The quotients are then 1/6,1/6, 2/9, 1/3, 1/3, 1/3, 1/3, 1/3. Finally, we get e_0=1 with weight Λ. Since some of the adjacent fractions are identical, this means that we do not need the entire sequence μ_8,μ_7,…,μ_1 but that, in writing a standard LS-representations, the μ_j corresponding to differences which are equal to zero should be omitted.  
 When we eliminate all j for which e_j=e_j+1, 
we are then left with μ_8,μ_6,μ_5,Λ as the weights, and quotients  1/6, 2/9, 1/3,1 as the fractions. Since we take difference between the fractions, the actual Littelman path is

    1/6μ_8, 1/18μ_6, 1/9μ_5,2/3Λ,

with the coefficients summing to 1 as they should.


We are dealing, in this paper, with the e-regular case, in which addable node are added from the top right down to the bottom left, as distinguished from the
 e-restricted case in <cit.> and <cit.> where the addable nodes are added from the bottom left to upper right.  In trying to reconstruct the multipartition in a non-recursive fashion from the Littelmann path in the form we have written it, all we have is the number of nodes of Y(λ) intersecting (Y(μ_j)-Y(μ_j-1)), not their location.  However, if they are fact located from the top down, then we could indeed reconstruct λ if we knew the reduced word, giving us the d_m,  and the fractions e_m correponding to the various μ_j, which would give us the c_m as d_me_m.  




§ THE CASE Λ=AΛ_0+BΛ_1


We consider the case e=2, with multicharge s=(0,0,…,0,1,1,…,1).   If there are a copies of 0 and b copies of 1, with level r=a+b, then when we fill in residues in the nodes of the Young diagram of a multipartition with this multicharge, there will be a of the  0-corner partitions followed by b of the
1-corner partitions.
   A partition in which the parts  alternate between odd and even will be called alternating.
 A multipartition for e=2 will be residue-homogeneous if 


  * Every nonempty partition is alternating.

  * All 0-corner partitions have the first line of the same parity, and the 1-corner partitions have the first line of the opposite parity. The last row of every non-zero partition, except possibly the last, is odd.







 A multipartition for e=2 will be strongly residue-homogeneous if 


  * It is residue homogeneous, and

  * For each partition λ^i after the first partition, if the top row has length n, the previous partition ends with a triangle of n-1 rows and columns if both have the same corner residue, and if it is the top partition of the  the 1-corner partitions, then the previous partition ends with a triangle of length n-2 if it has fewer rows than  λ^i, and of n if it has n or more rows.                                                                                                                                                                                                                                                                                       

 


We recall that, by the dual of Mathas' result  Prop. 4.9  in <cit.>, a multipartition is 2-regular if and only if  the length a(λ^t)  of the first row of a partition is  less that or equal to the number of rows  ℓ(λ^t-1) in the previous partition, except on the boundary between 0 and 1, where, for words beginning with s_1, it can be greater by 1.  If the top row of the first 1-corner partition is not greater by 1, we say that the multipartition is non-increasing.



A strongly residue-homogeneous multipartition is 2-regular.



We begin with two adjacent partitions λ^t-1 and λ^t in the multipartition which have the same corner residue, either 0 or 1. By the definition of residue homogeneous, the top rows have the same parity.  Letting n=a(λ^t), the length of the top row of the lower partition, then by the condition that the multipartition be strongly residue homogeneous, the upper partition ends in a triangle with at least n-1 rows.  Thus ℓ(λ^t-1) ≥ n-1.
Because λ^t-1 is a partition with no two rows identical, a(λ^t-1) ≥ℓ(λ^t-1). By the condition that the parities are the same, we get that ℓ(λ^t-1) ≥ n, which is the first condition needed for the dual Mathas result quoted above the lemma. At the boundary between 0-corner partitions and 1-corner partitions, we no longer have the parities equal, but we get as before that  ℓ(λ^t-1) ≥ n-1, and this is enough.


 For the case of e=2, there are only two families of reduced words.  The words with s_0 first on the right, i.e., … s_1s_0,  give the defect 0 weight whose multipartitions have the upper triangular partitions larger  will be denoted by   ψ^-_m, m>0. The words with s_1 on the right, i.e. … s_0s_1, give the defect 0 weights  ψ^+_m, m≥ 0 whose upper triangular partitions are smaller.
 All the 0-corner partitions have the same 
first row m, and all the 1-corner partitions all have the same length first row, which is either m-1 or m+1, respectively. When b=0, we will use ψ_m^- by default.



The floor and ceiling in the case r=1 are determined by the length  ℓ(λ) and  a(λ) of the first column and first row, respectively.  The situation for r>1 is considerably more complicated. 
In the case of general e for which we defined the standard LS-representation, the denominator d_m depended on the choice of reduced word. 

We now define d^±_n to be the number of rows in a defect 0 multipartition corresponding to a word beginning on the right  with s_1 or s_0, respectively, where n is the length of the first row.  Since all partitions with 0 in the corner have n rows, and all partitions with 1 in the corner have n ± 1 rows, we get

    d^±_n=an+b(n ± 1)=rn ± b=r(n ± 1)∓ a

The corresponding projection of the weight to the hub is 

    θ^±_2s=[d^±_2s+1,-d^±_2s]


    θ^±_2s+1=[-d^±_2s+1,d^±_2s+2]

In order to avoid writing every hub twice for the odd and even cases, we let ϕ be the operation of interchanging the 0 and 1 coordinates and then can write 

    θ^±_m=ϕ^m[d^±_m+1,-d^±_m]

Note that when a=1 and b=0, d_m^-=m.



 The segments are defined by putting together rows whose lengths drop by only one, with the following exception:
if the length of the first row of the partition equals the number ℓ  of rows in the previous partition, or, at the boundary between 0 and 1, equals ℓ± 1, then it does not start a new segment. The segment boundary is  the line between the last row of the old segment and the first of the new. For strongly residue-homogeneous multipartitions, the segments will correspond to the long paths of the corresponding unidirectional Littelmann path.





We number the segments from 1 to k, and for segment i  


  * We let v_i be the number of partitions before the end partition of the segment, where v_i=0 if the segment ends in the first partition.

  * Inside the last partition intersecting the segment, we let t_i be the number of  rows inside the partition down to the last row of the segment.

  * We let z_i be a Boolean parameter equal to 1 if the segment ends in a singleton and 0 if not. 

  * Let n_i be the length of the first row of the segment as before.  If the segment starts at the top of a 0-corner partition, then set n_i'=n_i, and at the top of a  1-corner partition, n_i'=n_i ∓ 1.  If it starts in the middle of  a 0-corner partition, set n_i'=n_i +t_i-1 and in the middle of a 1-corner partition, n_i'=n_i+t_i-1∓ 1.  



 Because the partitions are alternating and the first rows of 0-corner partitions all have the same parity opposite to that of the 1-corner partitions, all the n_i' will have the same parity.  We also have that 
n_1'>n_2'> …> n_k'. At each segment boundary, we extend the bottom row of segment i upward to the top of the partition, which it will hit at n_i', and this will be less than the point n_i-1' at which the segment i-1 will hit the top of the partition, since otherwise the two segment would have been combined into one.

The segment corresponds to  a number of adjacent rows in the multipartition with weight ψ_n_i'^- or  ψ_n_i'^+. If λ is a non-increasing partition, we will use “-” and if it is an increasing partition at the boundary between 0-corner and 1 corner partitions, we use ψ_n_i'^+.  The default, when the multipartition does not contain non-empty 1-corner partitions, will be “-”.



The proof of our main theorem, that a strongly homogeneous multipartitions has a unidirectional Littelmann path, involves a double induction.  The main induction is on the length n of the top row of the multipartition, but there will be a further internal induction on the segments.  In the course of this second induction we will need to know not only the lengths of various ladders, but also the lengths of the intersections of the ladders with the various segments. In order to calculate this precisely, we need more notation.



 Letting m be a number satisfying n_i+1'<m ≤ n_i',  we set v^m=v_i, the number of partitions above the partition containing the last row of the segment. 

Our aim is to find a formula for the  length of the m-ladder that contains the  mth-node in the top row of the multipartition.  The integer v^m will allow us to find the length of the intersection of the m-ladder with the partitions above the end partition, and now we must define a new quantity to measure the length of the part of the m-ladder in the last partition of the segment.  If the m-ladder ends on the last row of the segment, then this length is t_i, and particularly if  m=n_i',  the number of rows in the last partition intersection the segment.  If the m-ladder ends in the first column, then the length of its intersection is m or m ± 1, depending on whether we are in a 0-corner partition or a 1-corner partition.  In either case we get a number less than or equal to t_i.  Therefore we define



    t^m=
    min(t_i,m),    v_i < a
    min(t_i, m ± 1),    v_i ≥ a

for each such m, where we use the option “-" for non-increasing multipartitions.
We can combine the two cases as 

    min(t_i, m ± 1_v_i ≥ a).

 The integer t^m was defined to give exactly the number of nodes in the intersection of the ladder with the last partition.

 We can calculate d^±_m as above, and we can now give a algebraic formula for the lengths of the m-ladders. 


    c_m=
    
    v^m· m +t^m,   v^m ≤ a
    
    a · m +(v^m-a) · (m ± 1) +t^m    v^m>a


If we use the standard function x_+=max(x,0), then we can unite the two cases into a single formula

    c_m=v^mm±(v^m-a)_++t^m.


If this is in segment i, we get

    c_m=v_im ±(v_i-a)_++min(t_i, m ± 1_v_i ≥ a).


 The value of c_m will change in the middle of the segment i whenever v_i>0 and also if t^m depends on m.  We always have t^m=t_i when m=n_i', but if the segment ends in a singleton or if i=k and we pass to the seed part of the path, we can have  t^m<t_i.   
As in the introduction, we set

    e_m=c_m/d_m


In the  induction step of the main theorem, we will be required to prove that each segment corresponds to a straight path of integral length in the positive coordinate of the hub, whose length equals the number of addable nodes in that segment.  We will be required to express this integral  number in terms of various numbers associated with the multipartition, which we now define.


In order to deal with cases n_i+1'<m<n_i', we extend the Boolean parameter z_i from Def. <ref>  by setting z^m equal to 0 if the intersection of the segment with ψ^±_m does not  end with a singleton, and equal to 1 if it does.

To deal with the last segment, we let n'_k+1=0 if it ends in a singleton, i.e., z_k=1.  if the last row is longer than 1, we prepare the way for starting a new segment at the end of the current partition by setting z_k+1=1 if the last row is odd and  z_k+1=0 if the last row is even,   and set   n'_k+1=t_k ∓ 1_v_i ≥ a,n_k+2=0.   When m=n_k+1', let z^m =z_k+1.Finally, 
define

    I_m=c_m-c_m+1+v^m+z^m.



 We divide into cases:


  * If  m=n_i' for i ≤ k, then I_m=c_m-c_m+1+v^m+z^m is the number of addable rows in the segment i, with  

    I_n_i'   =(v_i-v_i-1)(n_i'+1) ± ((v_i-a)_+-(v_i-1-a)_+)
       +(t_i+z_i-min(t_i-1,n_i'+1 ± 1_v_i ≥ a))


  * If  n_i+1'<m<n_i', for 1 ≤ i ≤ k+1, then I_m=0. 

  * If m=n_k+1'>0, then I_m=z_k+1. 

 




If m=n_i' for i ≤ k, then t^m=t_i, v^m=v_i, z^m=z_i, and  c_m+1 belongs to segment i-1.  We then get 

    I_m   =c_m-c_m+1+v^m+z^m
       =(v_im ±(v_i-a)_++t_i)
       -(v_i-1(m+1) ±(v_i-1-a)_+)+min(t_i-1,m+1 ± 1_v_i-1≥ a))+v_i+z_i.
       =(v_i-v_i-1)(n_i'+1) ±((v_i-a)_±-(v_i-1-a)_+)
       +(t_i+z_i-min(t_i-1,n_i'+1 ± 1_v_i-1≥ a))





It remains to show that this is the number of addable nodes in segment i. All the addable nodes of segment i lie in the n_i'+1 ladder. The number of possible internal partitions of the segment is  v_i-v_i-1 .  This would give us 
(v_i-v_i-1)(n_i'+1) if they were all 0-corner partitions, but to adust for the possibility that some of the partitions are 1-corner partitions and are thus larger or smaller by 1, we must add an adjustment term ±((v_i-a)_+-(v_i-1-a)_+). Now from the last partition, there are an additional t_i+z_i addable nodes in the segment.  If the segment i-1 ends in middle of a partition, we have to remove t_i-1, since those nodes belong to the previous segment.  If the segment ends at a segment boundary, we need to remove only the nodes in the n_i'+1-ladder, of which there are n_i'+1 ± 1_v_i-1≥ a.

If n_i+1' <m<n_i', then c_m+1 belongs to  the same segment i, so c_m-c_m+1=-v_i-z_i, which means that I_m=0.

If  n_k+2' <m ≤ n_k+1, then we are not in a real segment, but in a virtual segment, which will only become a real segment if we decide to add the one addable node in the next row when z_k+1=1 .  We have  c_m-c_m+1=-v_k-z^m+1, so I_m=z^m-z^m+1.  If m<n_k+1' , then   z_m+1=z^m=1 so again I_m=0 in that case.

 If m=n_k+1'>0, then z^m+1=0 and z^m=z_k+1, and so I_m=z_k+1 and thus is either 0 or 1, depending on whether there is or is not an addable node.





The integer I_n_i' is the number of times we must operate by the appropriate f_ϵ in order to fill in the segment to the next level, and we are interested in having this as our denominator for the long paths.  

For e=2, Λ=3Λ_0+2Λ_1, the multipartition

 
    [(11,10,7,6,5,4,3,2,1),(7,6,5,4,3,2,1),(7,6,3,2,1),  (2,1), ∅]

has five segments, containing, 2, 7,9,3,2 rows where the respective segment pairs (v_i,t_i)  are 
(v_1,t_1)=(0,2), (v_2,t_2)=(0,7), (v_3,t_3)=(2,2),(v_4,t_4)=(2,5), (v_5,t_5)=(3,2).

Since c_12=0 and v^11=z^11=0, we get I_11=c_11=2. For the second segment, we have n_2'=9, so I_9=c_9-c_10+v^9+z_1=9-2+0+1=8. For the third segment, we use c_7=16, c_8=8, v^8=2, so I_7=16-8+2=10 .  The fourth segment has 3 rows, and we get  I_5=c_5-c_6+v^5+z_4=15-14+2+1=3. In the fifth and last segment we have by I_2=c_2-c_3+v^2+z_5=8-9+3+1=3. All the remaining I_m are zero. The last row ends in a singleton, so there is no virtual segment k+1.




 Every 2-regular non-increasing strongly residue-homogeneous multipartion λ can be built up from the empty partition by a sequence of operations  of the following two types:
 

  * (Widening) Adding every addable node down to the segment boundary of the last non-zero  segment, giving λ^*.

  * (Deepening) Adding every addable node down to the end of λ and then one singleton, in the same partition if the last row is odd and in the next partition if the last row is even and not at the boundary from 0-corner to 1 corner residues, to get λ̅ and then, if desired, adding singletons of the same corner  residue to some of the addable nodes in  previously empty partitions to get λ̃. A  deepening operation will always be followed by a widening or by a deepening which does not add a new partition with the same residue in the corner.




Induction on the number m  of nodes in the first line of the multipartition λ.   Assume that the lemma is true for every  strongly residue-homogeneous multipartition with first line less than m. If the last row  of the last non-zero segment  is  not a singleton, then removing  a  node from each row above the segment boundary is a reversible operation whose inverse is a widening, producing a strongly residue-homogeneous multipartition with first row m-1, to which we can apply the induction hypothesis. The second condition in the definition of  strongly residue homogeneous is stable because, though we reduce the triangles at the ends of each partition  except the last by removing one node from each row, we also reduce the first row of each partition by one node.

 If the last non-zero row is a singleton, then we have two cases.  If it is the last row of a larger partition, we apply a reversible operation whose inverse is a deepening, removing one node from each non-zero row. The last non-zero row will vanish. The number of rows in the triangular tails will be reduced by 1, but so will the first row of each partition.  We are reduced to a strongly residue-homogeneous multipartition with top row m-1 and we apply the induction hypothesis.

We will call the partition (1) a solitary partition. If the singleton is a solitary partition, then we must remove all the solitary partitions above it.  If it is in the 1-corner partitions, then this process must stop at or before the boundary between 0-corner and 1-corner partitions, because at that point the residue of a solitary partition changes from 1 to 0, and our multipartition is residue homogeneous. If it is a 0-corner partition, then if all the partitions are solitary, we have gotten back to the empty partitions.   We now consider two subcases.



  * Suppose that the corner of the upper solitary partition and the last non-solitary partition have the same corner residue. If the last row of this lowest non-solitary partition is a singleton, then we remove one node from each row and have a deepening.  If it is not a singleton, then by the definition of strongly
 residue-homogeneous, it must be odd and removing one node from each row  will give an even last row and we will have a deepening which adds partitions.


  * Suppose the corner of the  solitary partition has corner residue 1 and  the last non-solitary partition has corner residue 0.  Then by the definition of strongly residue-homogeneous the last 0-corner partition must end in a singleton, and we continue by the reverse of a deepening.





Our aim is to show that we have  long paths alternating with oscillating short paths. The lengths of the long path corresponding to segment i will correspond to the number of ϵ-addable nodes in the segment.  The hub of the long path corresponding to segment i is a multiple of θ_m, where m=n_i'. The positive coordinate of θ_m has value d_m+1 and, in a standard Littelmann path, the coefficient is e_m-e_m+1. We begin with a technical lemma, which, when combined with Lemma <ref>, will do most of the work needed to demonstrate unidirectionality.    


 Let λ be a 2-regular strongly residue-homogeneous multipartition with standard LS-representation. 
Let m be an integer such that   n_i+1'<m ≤ n_i'. The coefficient q^±_m=e_m-e_m+1 of the hub  θ^±_m in the Littelmann path can be written in the form 

    q_m^±=I_m/d^-_m+1+C_i/d^-_md^-_m+1

where I_m =0 unless m=n_i', in which case

    I_n_i'=(v_i-v_i-1)(n_i'+1)-((v_i-a)_+-(v_i-1-a)_+)+(t_i+z_i-min(t_i-1,n_i'+1-1_v_i-1≤ a))

and

    C_i=
    ∓ v_ib± r(v_i-a)_++rt_i,    z_i=0, i ≤ k, 
    ∓ (v_i+1)b± r(v_i-a)_+± r1_v_i ≥ a,    z_i=1,  or i=k+1



Before deriving the formulae, we note now that we will substitute d^±_m+1=d^±_m+r  in the numerator  but not in the denominator.  We will also use the formula d_m^±=rm ± b.
By definition,

    q^±_m=   e_m-e_m+1
    =   c_m/d^±_m-c_m+1/d^±_m+1
    
    =   c_md^±_m+1-c_m+1d^±_m/d^±_md^±_m+1.
    
    =   (c_m-c_m+1)d^±_m+rc_m/d^±_md^±_m+1.

Now  I_m=c_m-c_m+1+v^m+z^m is the numerator we want for d_m+1^±.  In order to get it, we must add and subtract  (v^m+z^m)d_m^±, getting

    q_m^±=   (c_m-c_m+1+v^m+z^m)d^±_m/d^±_md^±_m+1+-(v^m+z^m)d^±_m+rc_m/d^±_md^±_m+1
    
    =   I_m/d^±_m+1+-(v^m+z^m)(rm ± b)+rc_m/d^±_md^±_m+1,
    
    =   I_m/d^±_m+1+-v^mrm-z^m rm ∓ v^mb ∓ z^m b+r(v^mm+t^m±(v^m-a)_+)/d^±_md^±_m+1
    
    =   I_m/d^±_m+1+∓ v_ib± r(v_i-a)_+ ∓ z^mb+r(t^m-z^m m)/d^±_md^±_m+1,

where in the last step we cancelled two copies of  v^mrm  with opposing signs and substitute v_i for v^m.
We now divide into cases, since the values of z^m and t^m may depend on m.

Case 1:  z_i=0, i<k.

Then z^m=z_i=0 for all m satisfying n_i+1'<m ≤ n_i', and t^m=t_i, so by substitution we get 

    C_i=∓ v_ib± r(v_i-a)_++rt_i

Case 2:  z_i=1.

Then z^m=z_i=1 for all m satisfying n_i+1'<m ≤ n_i', and t^m=m ± 1_v_i ≥ a, where the additional 1 is added or subtracted only in the case v_i ≥ a.  Then t^m-z^mm=± 1_v_i ≥ a, so by substitution we get 

    C_i=∓ (v_i+1)b± r((v_i-a)_+ +1_v_i ≥ a)



Case 3:  i=k+1:   If m  ≤ n_k+1', we have z^m=1, t^m=m±  1_v_i ≥ a so then we get, as in Case 2, 


    C_i=∓ (v_i+1)b± r((v_i-a)_++1_v_i ≥ a)




 For i<k, or i=k, and either z_i=1 or  z_k=0 and m ± 1_v_i ≥ a> t_i we have 

    C_i=∓ (v_i+z_i)b± r(v_i-a)_++r((1-z_i)t_i ± z_i1_v_i ≥ a)),



In the listed cases,  the choice of formula depends only on the value of z_i, so we take a weighted average with weights z_i and 1-z_i.


The number C_i will be called the oscillator, and the important point is that there are two kind of oscillator. The first, called the widening oscillator, is for oscillating paths following a segment not ending in a singleton, or between the last segment and the transit path, while the second, which is independent of t_i, is only used for after segments which end in a singleton, where it is called a deepening oscillator  or for  the oscillating paths after the (possibly empty)  transit path connecting to the (possibly empty) tail, where it will be called the terminal oscillator.

When a=1 and b=0, so that we have a single partition, then C_i=t_i.  What makes the general case so much more complicated is that the addable nodes of segment i lie not only at the ends of rows, but also at boundaries between partitions. 
This segment boundary adjustment will occur frequently in our calculations, so denote it by s_i, where

    s_i=v_i+z_i-v_i-1-z_i-1
  

 For i ≤ k, we have 

    C_i-C_i-1=rI_n_i'-d_n_i'+1s_i,

For i=k+1, we have 

    C_k+1-C_k=-d_n_k+1',

If i=k+2, with w non-zero partitions and  y added partitions, then 

    C_k+2-C_k+1=
    
    yb,   w≤ a,
    
    -ya,    r>w>a,

Thus the terminal oscillator rises in steps of b from 0 to ab until the primary  tail a-w/aθ_0 is used up, then descends from ab to 0 is steps of a=d_1, in such a way that  the secondary tail, which  coincides with the oscillating path C_k+1/d_1d_2θ_1=r-w/d_2θ_1  in λ̅, becomes
C_k+2/d_1d_2θ_1=r-w-y/d_2θ_1 in  λ̃.


The right side of the equation  contains a large term equal to r(m+1)(v_i-v_i-1) which occurs twice with opposite signs and must be cancelled, so we will begin with the right side of the equation. 

    rI_m-d_m+1s_i   =r [(v_i-v_i-1)(m+1)-((v_i-a)_+-(v_i-1-a)_+) ]
       + [(t_i+z_i-min(t_i-1,m+1-1_v_i-1≥ a)]-(r(m+1)-b)s_i
       =r [-((v_i-a)_+-(v_i-1-a)_+) ]
       + [t_i+z_i-min(t_i-1,m+1-1_v_i-1≥ a)]-(r(m+1)(z_i-z_i-1))+bs_i
       =r  [-(v_i-a)_++(1-z_i)t_i+z_i1_v_i ≥ a ]+(v_i+z_i)b
       +r [(v_i-1-a)_+-(1-z_i-1)t_i-1 -z_i1_v_i-1≥ a ]-(v_i-1+z_i-1)b.
       =C_i-C_i-1


In the i=k+1 case, we have v_k=v_k+1, and t_k=n_k+1± -1_v_k+1≥ a, so

    C_k+1-C_k   =  [(v_k+1+1)b ± r(v_k+1-a)_+± r1_v_k+1≥ a ]
       +[ v_k b ± r(v_k-a)_++rt_k  ].
       =b± r1_v_k+1≥ a-rt_k 
       =b± r1_v_k+1≥ a-r(n_k+1± 1_v_k+1≥ a)
       =b-r(n_k+1')
       =-d_n_k+1'


If w is the number of non-empty partitions in λ, then from the formula for the second oscillator, substituting w for v_k+1, we have that the terminal oscillator C_k+1 is

    C_k+1=
    
    wb,   w≤ a,
    
    (r-w)a,    r>w>a,

The new terminal oscillator C_k+2 will be the same with w replaced by w+y, all within the 0-corner or 1-corner partitions, since we are adding the y solitary partitions with a single residue.





 Consider e=2, a=3, b=2. We calculate the q_m in this format for a sequence of multipartitions. The first has a transit path at θ_3, with d_3=13 and 17=C_1=C_2+d_3. The fourth has a transit path at 5, with d_5 =23 and 27=C_2= C_3+d_5. We will continue to have C_k+1=4 until we open a new partition and get terminal oscillator 6 or 3.


  * [(5,4,3,2,1)(5,4,3), ∅,∅,∅]

e: 8/23, 7/18, 6/13, 4/8, 2/3,1,

(9/28+17/28· 23)θ_5, 17/23· 18θ_4,  17/18· 13θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0

  * [(6,5,4,3,2,1)(6,5,4,1), ∅,∅,∅]

e: 9/28, 8/23, 8/18, 6/13, 4/8, 2/3,1,

(10/33+17/33· 28)θ_6, 17/28· 23θ_5,(2/23+4/23· 18)θ_4,  4/18· 13θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0

  * [(7,6,5,4,3,2,1)(7,6,5,2,1), ∅,∅,∅]

e: 10/33,9/28, 10/23, 8/18, 6/13, 4/8, 2/3,1,

(11/38+17/38· 33)θ_7, 17/33· 28θ_6,   (3/28+4/28· 23)θ_54/23· 18θ_4,   4/13 · 8θ_3, 4/13 · 8θ_2, 4/8· 3θ_1,1/3θ_0

  * [(8,7,6,5,4,3,2,1)(8,7,6,3,2), ∅,∅,∅]

e:  11/38,10/33,11/28, 10/23, 8/18, 6/13, 4/8, 2/3,1,

(12/43+17/43· 38)θ_8,17/38· 33θ_7,(2/33+ 27/33· 28)θ_6,  27/28· 23θ_5,4/23· 18θ_4,   4/13 · 8θ_3, 4/13 · 8θ_2

, 4/8· 3θ_1,1/3θ_0

  * λ= [(9,8,7,6,5,4,3,2,1)(9,8,7,4,3), (1),∅,∅]

e:12/43,  11/38,12/33,11/28, 10/23, 8/18, 6/13, 4/8, 3/3

(  13/48+17/48· 43)θ_9,17/43· 38θ_8,(2/38+27/38· 33)θ_7,27/33· 28θ_6,  (1/28+4/28· 23)θ_5,4/23· 18θ_4,   4/13 · 8θ_3, 4/13 · 8θ_2,

(1/8+1/8+6/8· 3)θ_1

  * λ^*= [(10,9,8,7,6,5,4,3,2,1)(10,9,8,5,4,1), (2),∅,∅]

e: 13/48,12/43,  13/38,12/33,12/28, 10/23, 8/18, 6/13, 5/8, 3/3

( 14/53+17/53· 48)θ_10, 17/48· 43θ_9,(2/43+27/43· 38)θ_8,27/38· 33θ_7,(2/33+4/33· 28)θ_6,  +4/28· 23θ_5,4/23· 18θ_4,   4/13 · 8θ_3, 

( 1/13+9/13 · 8)θ_2,
 (1/8+6/8· 3)θ_1

  * λ̅= [(10,9,8,7,6,5,4,3,2,1)(10,9,8,5,4,1), (2,1),∅,∅]

e: 13/48,12/43,  13/38,12/33,12/28, 10/23, 8/18, 6/13, 6/8, 3/3

( 14/53+17/53· 48)θ_10, 17/48· 43θ_9,(2/43+27/43· 38)θ_8,27/38· 33θ_7,(2/33+4/33· 28)θ_6,  +4/28· 23θ_5,4/23· 18θ_4,   4/13 · 8θ_3, 

( 3/13+6/13 · 8)θ_2,
 6/8· 3θ_1

  * λ̃= [(10,9,8,7,6,5,4,3,2,1)(10,9,8,5,4,1), (2,1),(1),∅]

e: 13/48,12/43,  13/38,12/33,12/28, 10/23, 8/18, 6/13, 7/8, 3/3

( 14/53+17/53· 48)θ_10, 17/48· 43θ_9,(2/43+27/43· 38)θ_8,27/38· 33θ_7,(2/33+4/33· 28)θ_6,  +4/28· 23θ_5,4/23· 18θ_4,   4/13 · 8θ_3,
 
( 5/13+3/13 · 8)θ_2,
 3/8· 3θ_1

Note the changes in the oscillators caused by transit paths, from 4 to 17, from 4 to 27 and, in λ^*, from 6 to 9.



Our aim is to prove that  every non-increasing strongly residue-homogeneous multipartition λ has a unidirectional  Littelmann path.  This Littelmann path will have a standard LS-representation corresponding to λ. The proof will proceed by induction. 
The assumption will imply that the Littlemann path consists of long paths separated by oscillating paths.   Given a strongly residue-homogeneous multipartition λ whose addable nodes are all of residue ϵ, we recall that we defined λ^* to be the multipartition obtained by adding ϵ-addable nodes down to the last non-zero row, and if the last row of λ is odd, we defined λ̅  to be the multipartition obtained by adding nodes of residue ϵ in place of all the ϵ-addable nodes down to the end of the last non-empty partition, including a singleton at the end. By applying the Littelmann algorithm for constructing f_ϵ, we will want to show that λ^*, and if possible  λ̅, both have standard Littelmann paths, and that if we continue to fill up empty partitions with singletons of residue ϵ, the new multipartition λ̃ also has standard Littelmann path.



 In rank e=2, for highest weight aΛ_0+bΛ_1, and a>0, any 2-regular non-increasing strongly residue-homogeneous multipartition  λ    has a Littelmann path π_λ  with  a standard LS-representation for λ which satisfies in addition that


  * If some partitions are empty, there is a primary  tail whose length in the 0-direction is the number of 0-corner ∅, or, if every 0-corner partition is non-empty, there is a secondary  tail whose length in the 1-direction is the number of 1-corner ∅ in λ. 

  * The Littelmann path is unidirectional.



 We are trying to prove this theorem only in the case of -, since, as we will show in the last section, it is not generally true in the case of +, so we will simplify the notation by letting d_m=d^-_m and θ_m=θ^-_m.


We will show by induction on the length of the top row n that the Littelmann path is standard for its multipartition. Using Lemmas 4.3 and <ref>, this will give us a unidirectional Littelmann path, where we will take the m_i to be the numbers n_i' for i ≤ k,
the integral length of the long path with index i will be I_n_i', the oscillators will be the C_i from Lemma <ref>. If the last segment, which has index  k, ends in a singleton there will be no transit path and m_k+1 will be the index of the floor, which will be 0 or 1 if there are empty partitions, but may be larger if there are no empty partitions. If the last segment does not end in a singleton, there will be a transit path. 

 Relying on Lemma <ref>, we assume that we have a λ with standard LS-representation which is unidirectional, and using the operations given there we will do either a widening to get λ^*, a deepening appending a singleton  to get λ̅,  or a deepening which adds solitary partitions (that is, with a single part of length 1) to get λ̃. Furthermore, our multipartitions can have several segments, so we must do an induction on the segments as well.  Our hypothesis on this inner induction is that all the segments previously treated consist of long paths with oscillators,  with coefficients as in a standard LS-representation.  

 For all segments i with i<k, the multipartitions and the Littelmann path for all three operations will be identical, so to condense the notation, we will take the multipartition λ and add the ϵ-addable nodes of that segment, of which there are  I_n_i', to get the new segment in λ^*.  We then perform the operation f_ϵ to the corresponding long path of π_λ, and check that the various invariants  e_m, m_i, and C_i of the segment i  match up to the new invariants  e_m^*, m_i^* and C_i^*. Then for i=k, we must take  e_m, I_m,  m_k, and C_k and use them to find the new  e_m^*, I_m^*, m_k^*, and  C_k^* in the case of widening, the new e̅_m, I̅_m,m̅_k, and C̅_k in the case of deepening, as well as the new ẽ_m, Ĩ_m, m̃_k and C̃_k' in the case of a deepening which adds extra solitary partitions.

In all cases, for 1 ≤ i ≤ k, the addition of all the ϵ- addable  nodes will add nodes to each row and  one node to each of the segment boundaries, so that we will have 
(n_i')^*=n_i'+1. In a segment which is entirely internal to a partition and does not end in a singleton,  the addable nodes will be added to existing rows, and we will have 
I_n_i'=I_(n_i')^*^*.  However, if the segment is not entirely internal to a partition, the number of addable nodes not added to previous rows is v_i+z_i from the bottom of the segment to the top of the multipartition, and similarly from the bottom of segment i-1 to the top of the multipartion we get  v_i-1+z_i-1 addable nodes at the boundaries between partitions. Thus in fact we must add the segment boundary adjustment s_i=(v_i+z_i)-(v_i-1+z_i-1) defined just before Corollary <ref>.

    I_(n_i')^*^*=I_n_i'+s_i

We divide into  four cases: for all i, continuing to widen or deepen the segment i, and for i=k, to widen after deepening, to deepen after widening and finally to adding  new solitary partitions.




  * 1 ≤  i  ≤ k, where if the segment ends in a singleton, we deepen and if does not end in a singleton, we widen.   For i<k, there is no difference between widening and deepening, and what we do is add all the addable nodes in  each segment.  Thus it will suffice to calculate  the new  e_m^*, m_i^*, I_m_i^* and  C_i^*, with some special treatment for the case i=k.  Each segment  has its own long path, of which the length is the number of addable rows in the segment. If we are at m=n_i', then we reflect the entire long path.  As in Lemma <ref>, this produces a new long path in the new direction, that of θ_m+1. By the induction hypothesis, λ has a standard LS representation, which depends on the sequence e_m=c_m/d_m, 1 ≤ m ≤ n. In order to prove that the widened multipartition λ^* has a standard LS-representation, we have to show that the new Littelmann path we get by the various reflections and translations given by Littelmann has an LS-representation given by a sequence  e_m^*=c_m^*/d_m, for 1 ≤ m ≤ n+ 1. Now in fact, c_m is only changed where nodes are added in the m-ladder.  For each segment i, the nodes are added in one particular ladder, that corresponding to n_i'+1. Thus for all remaining m in segment i, we have c_m=c_m^*, which means that for all q_m which do not involve n_i'+1, q_m^*=q_m. Furthermore, since we already showed that I_n_i' is the number of addable nodes in segment i, we have, for m=n_i', that  

    c_m+1^*=c_m+1+I_m


We now calculate the new q_m^* and  q_m+1^*.

    q^*_m=   e_m^*-e_m+1^*
    =   c_m^*/d_m-c_m+1^*/d_m+1
    
    =   c_m/d_m-c_m+1+I_m/d_m+1
    
    =   q_m-I_m/d_m+1
    
    =    (I_m/d_m+1+C_i/d_md_m+1 )-I_m/d_m+1
    
    =   C_i/d_md_m+1

This is the same oscillator that occurs in q_m-1, so we still have an oscillating part,  the new oscillator C_i^* is identical to C_i and I_m^* becomes 0 as we expect in the oscillating part.

We now calculate the coefficicient of θ_m+1 in the new Littelmann path obtained by reflecting the long path.  The  part which is reflected is the subpath I_m/d_m+1θ_m, so after I_m reflections it becomes I_m/d_m+1θ_m+1.  It joins the small oscillating part from the previous segment,  C_i-1/d_m+1d_m+2θ_m+1 so that the coefficient of θ_m+1 will be I_m/d_m+1+C_i-1/d_m+1d_m+2.



    q^*_m+1=   e_m+1^*-e_m+2^*
    =   c_m+1^*/d_m+1-c_m+2^*/d_m+2
    
    =   c_m+1+I_m/d_m+1-c_m+2/d_m+2
    
    =   I_m/d_m+1+q_m+1
    
    =   I_m/d_m+1+C_i-1/d_m+1d_m+2




 The number of addable nodes in the new segment should be the number of addable nodes in the segment i of λ plus the number of s_i of partition boundaries in the  segment, which is s_i=v_i-v_i-1+z_i-z_i-1. so that 

    I_m+1^*=I_m+s_i

We now rewrite q_m+1^* in terms of the invariants of λ^*, where at the end we apply Corollary <ref> and then the result we just proved that C_i=C_i^*.


    q^*_m+1=   I_m/d_m+1+C_i-1/d_m+1d_m+2
    
    =   I_m/d_m+2+C_i-1+(d_m+2-d_m+1)I_m/d_m+1d_m+2
    
    =   I_m+s_i/d_m+2+C_i-1+(d_m+2-d_m+1)I_m-d_m+1s_i/d_m+1d_m+2
    
    =   I_m+1^*/d_m+2+C_i-1+rI_m-d_m+1s_i/d_m+1d_m+2
    
    =   I_m+1^*/d_m+2+C_i/d_m+1d_m+2
    
    =   I_m+1^*/d_m+2+C_i^*/d_m+1d_m+2



 The little left-over piece at the end after we reflected most of the long path of segment i will be C_i/d_md_m+1θ_m.  Whereas the oscillating paths used to pair in a way which always returned them to the same integer value of the positive coordinate, they will now pair in a way which always returns them to the same integer value of the other coordinate.  If i<k, then the last piece will now join the next long path. For the case of deepening a segment  with i=k ending in a singleton, the entire calculation is valid if we just replace  q_m^* with q̅_m and similarly for m+1.

Note that if i=1, there was no previous small path, but in that case we also have C_i-1=0.
In summary, the long path of segment i has shifted up one index from m to m+1 as we passed from λ to λ^*.


  * i=k, where the segment ends in a singleton and we want to widen.  The case of i=k is considerably more complicated than the case of i<k because our induction must take into account five separate sections of the Littelmann path, which we now list in order going away from the origin:  The long path of segment k, the oscillating part of segment k, the possibly empty transit path, the possibly empty terminal oscillator and the possibly empty tail.  The transit path can only be used as a seed to start a new segment when the last row is odd.

 We are assuming that λ  has a standard LS-representation and is unidirectional.
We now take as our inner induction hypothesis that i=k, and  that segment k ends in a singleton. We wish  to do a widening and to compute the new oscillator.  Let m=n_k'.
We already showed in the previous section of this proof that if we were to do a deepening to get λ̅, adding all I_m addable nodes, we would get a new long path  with standard LS-representation, where

    q̅_m=C_k/d_md_m+1


    q̅_m+1=I_m/d_m+1+C_k/d_m+1d_m+2
 
  Instead of adding all the I_n addable nodes, we add only I_n-z_k  addable nodes.  In terms of the Littelmann path, this leaves a path of length  1 in the positive direction which will be the transit path and a continuation with the same terminal oscillator C_k as before.

    ( z_k/d_m+1+C_k/d_md_m+1 )θ_m=  ( C_k+z_kd_m/d_md_m+1 )θ_m

Now we check that this widening will still give a standard LS-representation with  C_k^*= C_k+z_kd_m as oscillator of segment k.  We have   c_m+1^*=c_m+1+I_m-z_k and c_ℓ=c_ℓ^* for  all other c_ℓ in the segment, so we need check only q_ℓ for ℓ=m, m+1.


    q^*_m=   e_m^*-e_m+1^*
    =   c_m^*/d_m-c_m+1^*/d_m+1
    
    =   c_m/d_m-c_m+1+I_m-z_k/d_m+1
    
    =   q_m-I_m-z_k/d_m+1
    
    =    (I_m/d_m+1+C_k/d_md_m+1 )-I_m-z_k/d_m+1
    
    =   C_k+z_kd_m/d_md_m+1


This is precisely the formula for the oscillator C_k^* in Lemma <ref>, where now z_i^*=0, and the path after widening is still unidirectional.  Thus we have a short path which is ambiguous:  it can either be thought of as the length 1 transit path with a continuation which belongs to the terminal ocillator, or as the first step in a new oscillating part. With m=n_k',

    q^*_m+1=   e_m+1^*-e_m+2^*
    =   c_m+1^*/d_m+1-c_m+2^*/d_m+2
    
    =   c_m+1+I_m-z_k/d_m+1-c_m+2/d_m+2
    
    =   I_m-z_k/d_m+1+q_m+1
    
    =   I_m-z_k/d_m+1+C_k-1/d_m+1d_m+2
    
    =   I_m-z_k/d_m+2+(d_m+2-d_m+1)(I_m-z_k)+C_k-1/d_m+1d_m+2
    
    =   I_m-z_k+s_k^*/d_m+2+r(I_m-z_k)-d_m+1s_k^*+C_k-1/d_m+1d_m+2
    
    =   I_m+1^*/d_m+2+C_k^*/d_m+1d_m+2

where the last step is a result of Lemma <ref> applied to λ^*,
 
    C_k^*=rI_n_k'^*^*-d_n_k'^*+1s_k^*+C_k-1^*,

using the fact that C_k-1=C_k-1^*  since k-1<k, that I_m+1^*=I_m+s_k-2z_k, that s_k^*=s_k-z_k   and that d_m+2 = d_m+1+r.
Further widening, as we saw in the first part of the proof, will move the long paths up by one, and will add new short paths with the same oscillator.  The position of the transit path will be unaffected, so we will have m_k+1=m_k+1^*and the section from the transit path to the tail will only be translated, so it will memain unchanged.



  * i=k.  Deepening after widening without adding a new partition.   The partition ended in a singleton when we first started to widen at m=m_k+1 and the transit path has a hub which is a multiple of θ_m_k+1. 

Thus, after widening to get λ^*, if the last row is odd  we can continue one more step and reflect the part of length 1 in the transit path to get λ̅.  This becomes the new long path of segment k+1. There will not be a transit path in the new seed part until after the next widening. The tail and the oscillating part before it remain as they were, so the new path is unidirectional.  It remains to show that the new path has a standard LS-representation.  The ladders c̅_m are identical to those of c_m^*, until we come to m=n_k+1'. The last row being odd, this is not the beginning of a segment for ℓ=m,m+1, so we have c_ℓ^*=c_ℓ.  In λ̅, we add a new node in the  m+1-ladder, so that   c̅_m+1= c_m+1+1.  We have already checked the LS-representation for ladders outside of m for λ^*, so now we check q̅_m, where I_m, the number of addable nodes in the new segment k+1,   is 2.


    q̅_m=   c_m/d_m-c_m+1+1/d_m+1
    
    =   q_m-1/d_m+1
    
    =    (1/d_m+1+C_k+1/d_md_m+1 )-1/d_m+1
    
    =    (C_k+1/d_md_m+1 )


    q̅_m+1=   c_m+1+1/d_m+1-c_m+2/d_m+2
    
    =   q_m+1+1/d_m+1
    
    =    (1/d_m+1+C_k/d_m+1d_m+2 )
    
    =    (d_m+2/d_m+1d_m+2+d_m+C_k+1/d_m+1d_m+2 )
    
    =    (2/d_m+2+C_k+1/d_m+1d_m+2 )


If the last row is even, we must either widen again or proceed to the next possibility, adding a new partition.



  * i=k.  Deepening to add new partitions. The new partitions must come from the tail.  In order to get to the tail, we must add all addable nodes between the last long path and the tail. There will be exactly one such addable node if there is a transit path, and none otherwise. If there is no transit path, then the solitary partitions will open a new segment and the addable node will belong to segment k.  If there is a transit path, and the addable node belongs to the same ladder as the solitary partitions, then the addable node will belong to the same segment as the new solitary partition. Under all other conditions, we will have to add two segments.  In order to treat all cases at once, if there is a transit path, we will call the segment with the solitary partitions k+2, and, when we finish deepening, rename it as k+1 if necessary.

It is only possible to add y solitary partitions is there are empty partitions, which is to say, if there is a tail,  which will be  as always at θ_m for 
m= 1_v_i-1≥ a.  Only c_m+1 will be changed by the addition of these addable nodes. We have already shown that λ̅ has a standard LS-representation and is unidirectional,  so as in the earlier parts of the proof, it only necessary to check q_m and q_m+1, and see that they correspond to the Littelmann path obtained by reflecting a piece of the tail of length y in the positive coordinate. 
It is also true that e_m=1.  This is by definition if m=0, and we can only have m=1 if c_1=d_1=a. We have c̃_m+1=c_m+1+y, and for the remaining relevant indices u we have c̃_u=c_u.


    q̃_m=   ẽ_m-ẽ_m+1
    =   c̃_m/d_m-c̃_m+1/d_m+1
    
    =   1-c_m+1+y/d_m+1
    
    =   d_m+1-c_m+1-y/d_m+1


which gives the new tail, possibly zero, and coincides with the tail calculated in Lemma <ref>.  Now, in order to find the new terminal oscillator, we calculate 



    q̃_m+1=   c_m+1+y/d_m+1-c_m+2/d_m+1d_m+2
    
    =   q_m+1+y/d_m+1
    
    =   q_m+1+2y/d_m+2+y(d_m+2-2d_m+1)/d_m+1d_m+2
    
    =   q_m+1+2y/d_m+2+y(-d_m)/d_m+1d_m+2


where the added piece  2y/d_m+2 gives two addable nodes for each of the added solitary partitions. If m=1,  since d_1=a, 
then the quantity y(-a)/d_m+1d_m+2 is precisely the difference C_k+2-C_k+1 from Lemma <ref>. In the case of m=0, 
then d_0=-b and again the quantity y(b)/d_m+1d_m+2 is the difference C_k+2-C_k+1 from Lemma <ref>.






To illustrate the theorem, we give  case where the partition crosses the boundary between the 0-corner and 1-corner partitions.
 Consider e=2, a=2, b=3. We calculate the q_m in this format for a sequence of multipartitions. The second has a transit path at θ_2, with d_2=7 and 13=C_1=C_2+d_2. The first four multipartitions have tails; the last does not, but it does have three long paths, and terminal oscillator 0.


  * [(2,1),(2,1),∅,∅,∅]

e:  4/7, 2/2,

(6/12+ 6/12 · 7)θ_2, 6/7· 2θ_1

  * [(3,2,1),(3,2),∅,∅,∅]

e:  5/12, 4/7, 2/2,

(6/17+ 13/17· 12)θ_3, 13/12 · 7θ_2, 6/7· 2θ_1

  * [(4,3,2,1),(4,3),(1),(1),∅]

e:  6/17, 5/12, 6/7, 2/2,

(7/22+ 13/22· 17)θ_4, 13/17· 12θ_3, (5/12+ 2/12 · 7)θ_2, 2/7· 2θ_1

  * [(5,4,3,2,1)(5,4,1), (2,1),(2,1),∅]

e: 7/22, 6/17, 10/12, 6/7, 2/2,

(8/27+13/27· 22)θ_5, 13/22· 17θ_4, (8/17+ 2/17· 12)θ_3, 2/12 · 7θ_2, 2/7· 2θ_1

  * [(6,5,4,3,2,1)(6,5,2,1), (3,2,1),(3,2,1),(1)]

e: 8/27, 7/22, 14/17, 10/12, 7/7,

(9/32+13/32· 27)θ_6, 13/27· 22θ_5,( 11/ 22+ 2/22· 17)θ_4,  2/17· 12θ_3, (2/12+0/12 · 7)θ_2.








§ MOTIVATION AND OPEN PROBLEMS




To motivate the definition of the residue-homogeneous multipartitions, we use the block reduced crystal graph from <cit.>, further  developed in <cit.>, in which the vertices are the weights 
Λ-α in P(Λ), where two weights connected by an edge if we can find a basis element from the weight space of each such that they are connected by f_i for some i. In Figure 1, we have drawn the case 
of Λ=2Λ_0+Λ_1, truncated at degree 17. Vertices on the same horizontal line correspond to multipartitions of fixed degree n. The diagonal edges down to the left, which are called 0-strings,  represent action by f_0 and the edges going down to the right, called 1-strings,  represent f_1.  The defects can all be determined in this case from the fact that acting on a string  by a simple reflection preserves defect, the highest weight element has defect 0, adding the null root δ adds the level r=3 to the defect, and (Λ-α_0)=1.  The number of multipartitions corresponding to a vertex is fixed for each defect.  In this case there is one multipartition for each vertex of defect 0 or 1, there are no vertices of defect 2, and there are 2 multipartitions for a vertex of defect 3.  We have written in the multipartitions for the external vertices of defect 0,1 and 3. 
We originally began to study the unidirectional Littelmann path in the context of external vertices.

It follows from the work of Scopes <cit.> and generalizations by Ariki and Koike <cit.>, Kleshchev and Brundan <cit.>, that if we have a weight all of whose crystal elements are external, then acting on the block of the cyclotomic Hecke algebra  by the Weyl group in the direction increasing the degree will produce a Morita equivalence. For every defect, there is a degree after which all weights of that defect are external, so the external vertices are a sort of limit case.  The set of strongly residue homogeneous multipartitions contained almost all of the external vertices, as well as a number of other vertices which are definitely not external. We defined  strongly residue homogeneous multipartitions here only for the case e=2, though it is possible to make an extension to larger ranks. We also consider unidirectional Littelmann paths only for defect 0 weights for which the reflection multiplying Λ is s_0, the reflection through the 0-th simple root α_0. Since we are in the case e=2, the dominant integral weight Λ is of the form Λ=aΛ_0+bΛ_1 and we set r=a+b, which in type A is the level.  



A vertex of the crystal  for which all multipartitions are residue-homogeneous multipartitions is external.



Let λ be a residue-homogeneous multipartition.  Since each partition is alternating, the rows in a partition all end with the same residue, and by the conditions on the parity of the first rows, this residue is the same for each partition.  Let 1-ϵ be the common residue at the end of all non-zero rows, so that the addable nodes at the ends are all of residue ϵ.   Since there are no ϵ removable rows, we must have that e_ϵ is zero.


The multipartitions labelling external vertices in Figure 1 are all strongly residue-homogeneous. Among them, the multipartition [(3),(1),(2,1)] is  strongly residue-homogeneous  because a singleton not at the boundary between 0- and 1-corner partitions makes no restriction on the previous partition. 

The main theorem is severely limited, in that it only applies to the case e=2 and even there, only to the non-increasing partitions.  We can in  fact get some results for e=3 and perhaps higher, but we must restrict ourselves to sequences of defect 0 weights which are periodic, that is to say, generated by words which are periodic n the Weyl generators.  The case of 3-periodic words is addressed in the original Ph.D. thesis, <cit.>. We again get long paths and short oscillating sections between them.  

To move in the other direction, from a unidirectional Littelmann path to its multipartition is harder, since the   I_m and the C_i are nonlinear functions of the multipartition parameters v_i,  t_i, and z_i. However, the nonlinear function, as we showed in Lemma <ref>, is a two part piecewise-linear function.  Furthermore, the same lemma allows us to determine the s_i, since, in the definition of unidirectional, we specify the C_i and 
I_n_i'. In our two main examples, we used  two distinct primes as a and b, which made it easy to identify the terminal oscillators, as small multiples of a or b, while the widening oscillators were much larger.


In order to explain what interferes with getting a standard LS-representation in the “+” case, where the first 1-corner partition has first row larger than the number of rows in the previous partition, as in [(1),(1),(2,1)], we look an easy case




 A pseudo-floor of λ is a multipartition which begins like a defect 0 multipartition with weight  ψ^± _ℓ, but is  truncated  after w partitions. 


 If it is increasing at the bundary between 0-corner and 1-corner partition, it will not have a standard LS-representation.
Suppose we have a pseudo-floor λ and do a deepening to λ̅ We need to show that e̅_n ≥e̅_n+1.  This is equivalent to showing the q_n≥ 0 . Since n<n_1' =n+1, we know from Lemma <ref> for λ̅ that I_n=0.  Thus we need to determine if C_1≥ 0.  We have i=1=k since there is only one segment and z_i=1 because we have a pseudo-floor, so we are in case the second case in Lemma <ref>.

    C_1= ∓(v_1+1)b± r((v_1-a)_++1_v_1 ≥ a)

In the + case, a<w<r.  We then have (v_1-a) ≥ 0, so we can drop the +.  Substituting
v_1+1=w and using b=(r-w)+(w-a), we have 

    C_1= -wb+ r(w-a)=(r-w)(w-a-w)=-(r-w)a<0.


Thus only the “-” case will give us an LS-representation.  Since the formulae for a pseudo-floor are simple, we compute both the oscillator and the tail:


    C_1=
    
    wb,   w≤ a,
    
    (r-w)a,    r>w>a,



Also for a pseudo-floor, we can calculate the hub of the tail


    hub(T)=
    a-w/aθ_0,   w< a,
    r-w/bθ_1,    r>w ≥ a


 Let Λ=2Λ_0+Λ_1 and let λ be the multipartition [(5,2,1),(1),∅]. 




  * λ= [(5,2,1),(1),∅]

e: 1/14,1/11,3/8,2/5,2/2

(1/17+ 3/17· 14)θ_5, 3/14 · 11θ_4, (3/11+ 1/11· 8)θ_3, 1/8 · 5θ_2, 4+2/5· 2θ_1

  * λ^*=[(6,3,2,1),(2),∅]

e: 1/17,1/14,4/11,3/8,3/5,2/2)
(1/20+ 3/20· 17)θ_6, 3/17· 14θ_5 ,(4/14+ 1/14 · 11)θ_4, 1/11· 8θ_3, (1/8+ 4/8 · 5)θ_2, 4/5· 2θ_1

  * λ̅=[(6,3,2,1),(2,1),∅]

e: 1/17,1/14,4/11,3/8,4/5,2/2,

(1/20+ 3/20· 17)θ_6, 3/17· 14θ_5 ,(4/14+ 1/14 · 11)θ_4, 1/11· 8θ_3, (3/8+ 2/8 · 5)θ_2, 2/5· 2θ_1

  * λ̃=[(6,3,2,1),(2,1),(1)]

e: 1/17,1/14,4/11,3/8,5/5,

(1/20+ 3/20· 17)θ_6, 3/17· 14θ_5 ,(4/14+ 1/14 · 11)θ_4, 1/11· 8θ_3, (5/8+ 0/8 · 5)θ_2.


In Figure 2, we draw the hubs of the Littelmann paths in the case of deepening.







There are three long paths, but in the last path there is a piece of 1-length b=1, which is actually the tail, and another piece of  1-length 2 which will be needed if we wish to add two nodes to the partitions (1).  If we choose to widen, then we get λ^*=[(6,3,2,1),(2),∅] in which only the first third of the last long path is reflected. If we choose to continue deepening to λ̅, without opening another partition, then we will reflect the second third get a multipartition  λ̃=[(6,3,2,1),(2,1),∅] in which the tail is obvious.
Finally, if we continue to fill in the last non-zero partition, the tail will straighten out in the direction of the last long path, giving the Littelmann path of 
λ̃.


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