LISTA DE REFERENCIAS

We quote results from the appendix of [PW]. There weight spaces of Demazure submod- ules of irreducible representations ofsln(_C) were described with semistandard tableaux with values from [n]. The axis basis, positive roots, and simple reflections were chosen accordingly. Here we consider irreducible representations of sln+1(C). Hence our tableaux contain val-

ues from [0, n], not [n]. We also use reverse semistandard tableaux instead of semistandard tableaux because they more naturally give rise to the Hillman-Grassl build-up viewpoint of this thesis. Therefore, when quoting from the appendix of [PW] we need to make those two changes. We choose the Cartan subalgebraHto be the subspace ofsln+1(C) consisting of the

diagonal matrices. For n≥ i≥0, we define φi ∈ H∗ to be the linear function that extracts the coefficient of the elementary matrixEn−i,n−i. These n+ 1 linearly dependent functionals are used to describe weights. In [PW], the axis basis was chosen to be the basis of linear functions {φi}ni=0 such that φi extracts the coefficient of the matrix Ei,i. For n ≥i ≥0, we haveφn−i =φi. Forn ≥i≥1, we define the positive simple roots to beαi =φi−φi−1. Then

the set Φ+ of positive roots is {φi−φj|n ≥ i > j ≥ 0} and the set Φ− of negative roots is

{−φi+φj|n ≥i > j≥0}. Here the Borel subalgebraB is the subalgebra of trace free upper triangular matrices. For n≥b ≥1, the fundamental weights are ωb =φn+φn−1 +. . .+φb. In the axis basis of H∗, the fundamental weight ωb can be depicted by the (n + 1)-tuple

(1,1, . . . ,1,0,0, . . . ,0) which contains n+ 1−b 1’s. Letλ =

n

X

i=1

aiωi be a dominant integral weight, whereai ∈Nfor 1≤i≤n. We produce an n-partition λ from this Lie dominant integral weight λ. This is the n-partition whose shape consists of ai columns of length n+ 1−i for 1≤i≤n. Recall from Section 2.1 that Q(λ) = {q1, q2, . . . , qk} is equal to the set of column lengths of the n-partition λ. Take for example the adjoint representation, which has highest weight λ =

n

X

i=1

ωi. The n-partition λ produced from the highest weight λ is the “staircase” shape which has one column of length i for n ≥ i ≥1. We then have Q(λ) = [n]. Given a dominant integral weight λ, the weights-with-multiplicities of the irreducible representation Vλ of sln+1(C) can be described

by the reverse semistandard tableaux on the shape λ with values from [0, n]. In [PW], the weight described by a semistandard tableau T is equal to

n

X

i=0

ciφi, where ci is the number of times i appears in T. For a reverse semistandard tableau T, the weight described by T is equal to

n

X

i=0

ciφi, where ci is the number of times i appears in T. Since φn−i = φi, to obtain our (n+ 1)-reverse semistandard tableaux from (n + 1)-semistandard tableaux, we subtract the values in the tableaux entrywise fromn. Given an irreducible representationVλ of sln+1(C), the highest weight λ is depicted by the semistandard tableau with the smallest

possible values. When we subtract all the values of the tableau fromn, the highest weightλ is depicted by the reverse semistandard tableau with the largest possible values. It can also be seen that this tableau describes the highest weight directly from the definitions without needing to subtract values from a semistandard tableau.

Fix a dominant integral weight λ. For n ≥ i ≥ 1, the simple reflection si permutes φi and φi−1. These simple reflections generate the Weyl group W = Sn+1. To precisely

index the Demazure submodules of Vλ, first set J := Jλ := {i ∈ [n] : si.λ = λ}. There is one distinct Demazure module for each coset wWJ in the set of cosets WJ := W/WJ. Each such coset has a unique minimal length representative inW; let Wλ _{denote the set of} these representatives. For some t ≥ 1 and i1, i2, . . . , it ∈ [n], we can write w ∈ Wλ as the

product of simple reflectionssit. . . si2si1. For w ∈W

λ_{, the set Φ(w) is defined to be the set} Φ+_∩w(Φ−_).

Fix w ∈ Wλ_{. Here we describe how to produce an ordered Q-partition ρ from w. Let} n≥i≥0. Defineρibyφρi :=w(φi). Setρ:= (ρn, ρn−1, . . . , ρ1, ρ0). From the following it can

be seen thatρ becomes the standard form of an ordered Q-partition: For our combinatorial model ofw, the simple reflections in the productw=sit. . . si2si1 act by value on the (n+ 1)-

tuple (n, n−1, . . . ,2,1,0). The positions of the values of this (n + 1)-tuple are indexed decreasing from n to 0 from left to right. For n ≥ i ≥ 1, the simple reflection si permutes the values iand i−1. The identity e∈Wλ produces the (n+ 1)-tuple (n, n−1, . . . ,2,1,0). Letλ be the n-partition produced from the weight λ and letQ(λ) = {q1, q2, . . . , qk}. Recall that we defined q0 := 0 and qk+1 := n + 1. For 1 ≤ r ≤ k, we place a semicolon into ρ

between positions n+ 1−qr and n−qr:

ρ:= (ρn, ρn−1, . . . , ρn+1−q1;ρn−q1, . . . , ρn+1−q2;ρn−q2, . . .;. . .;ρn−qk, . . . , ρ0).

This separates the set of positions into k+ 1 carrels. The rth _{carrel from the left consists} of the positions {n−qr−1, n−1−qr−1, . . . , n+ 1−qr}. The size of the rth carrel is equal to qr −qr−1. Placing these semicolons also separates the set of values into k+ 1 cohorts

corresponding to the k+ 1 carrels. When we mod out by the parabolic subgroup WJ, the positions within each of thek+ 1 carrels become indistinguishable. Thus the shortest length coset representative w ∈Wλ produces an (n+ 1)-tuple wherein the values within a cohort decrease from left to right. The resulting (n+ 1)-tuple is the standard form of an ordered Q-partitionsρ. It can be seen that every orderedQ-partition is produced once in this fashion. In the Section 1.1 generic statement of the Kumar-Peterson identity, the adjusted De- mazure charactersy−wmλ_D

mλ(w;y) were written in terms of a variabley. We used theyvari- able to denote a generic coordinatization of these adjusted characters. Now use the variables xi to coordinatize these adjusted characters with respect to the axis basis {φi}n

i=0: Here we

setxi := exp(φi), the formal exponential ofφi. We use the variables zi to coordinatize these adjusted characters with respect to the simple root basis{αi}ni=1. Here we havezi := exp(αi). We now have two ways to relate the sets of variablesxiandzi. Sinceαi =φi−φi−1, for our Lie

theoretic definition of these variables, we have zi = exp(αi) = exp(φi) exp−1(φi−1) =x−i−11xi . Note that this agrees with our combinatorial definition of zi :=x−i−11xi made in Section 2.7. We refer to the formal exponentialzi of the simple rootαias thez-weight of this simple root. We similarly refer to zizi+1. . . zj as the z-weight of the positive root αi +αi+1+. . .+αj. The term “weight” here is combinatorial terminology and does not refer to the Lie theoretic definition of weight.