Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 09 April 2012


An-1 2 3 n-1 n A -2 -1 0 1 2 An-1 0 2 3 n-1 n

Let I = { {1,2,...,l-1}, in type   Al-1, , in type   A, /l, in type   Al-1(1). } The elements of I index the (isomorphism classes) of simple representations of the quiver.

Consider a sheet of graph paper with diagonals indexed by . The content c(b) of a box b on this sheet of graph paper is c(b) = the diagonal number of the box b. Let [i;d) = [i,i+d-1] = (d;i+d-1] = i i+d-1 denote a sequence of boxes in a row which has length d with the leftmost box of content i and the rightmost box of content i+d-1. The set of segments is R+ = { { [i;d)  |  iI,  1dl-i }, in type   Al-1, { [i;d)  |  iI,  d0 }, in type   A, { [i;d)  |  iI,  d/l }, in type   Al-1(1). } The elements of R+ index the (isomorphism classes) of indecomposable (nilpotent) representations of the quiver.

A multisegment is a (unordered) collection of segments, i.e. an elements of B˜() = αR+ 0α. For example 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 = 2[3;5) + [5;3) + [1;5) + [3;3) (MS 1) (the numbers in the boxes in the picture are the contents of the boxes).

A multisegment is aperiodic if it does not contain [0;d) + [1;d) ++ [l-1;d), for any   d>0. Pictorially, a multisegment is aperiodic if it does not contain a box of height l. Let B() = {aperiodic multisegments}. In types Al-1 and A, B() = B˜(). The elements of B() index the isomorphism classes of nilpotent representations of the quiver.

The partial order

Consider an (infinite) sheet of graph paper which has its diagonals labeled consecutively by ...,-2,-1, 0,1,2,.... The content c(b) of a box b on this sheet of graph paper is c(b) = the diagonal number of the box   b. A segment is a row of boxes on a sheet of graph paper with diagonals indexed by .

Consider a graph paper with diagonals indexed by . A segment is a sequence of boxes [i,j] = i j in a row with the leftmost box of content i and the rightmost box of content j. A multisegment is a (unordered) collection of segments. For example 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 has λ([3,7]) = 2,  λ([5,7]) = 1,  λ([1,5]) = 1,  λ([3,5]) = 1,  and λ([i,j]) = 0 for all other segments [i,j] (the numbers in the boxes in the picture are the contents of the boxes). Alternatively a multisegment λ can be viewed as a function λ: {segments} 0 where λ([i,j]) = (#   of rows   [i,j]   in   λ). The set of segments is ordered by inclusion. Define λ([i,j]) = [r,s][i,j] λ([r,s]). (MS 2) Then λ([i,j]) = λ([ i-1,j+1 ]) - λ([ i-1,j ]) - λ([ i,j+1 ]) + λ([ i,j ])

and so the multisegment λ can be specified by the numbers λ( [i,j] ). Note that λ([i]) = (#  of boxes in   λ   in diagonal  i). Define a partial order on multisegments by λμ if λ([i,j]) μ([i,j]) for all segments   [i,j].

If [b,c] [a,d] are segments define a degeneration R [b,c], [a,d] : {multisegments} {multisegments} by R [b,c], [a,d] λ([ a,d ]) = λ([ a,d ]) -1, R [b,c], [a,d] λ([ b,d ]) = λ([ b,d ]) +1, R [b,c], [a,d] λ([ a,c ]) = λ([ a,c ]) +1, R [b,c], [a,d] λ([ b,c ]) = λ([ b,c ]) -1, R [b,c], [a,d] λ([ i,j ]) = λ([ i,j ]), if   [i,j] [a,d], [a,c], [b,d], [b,c]. The degeneration R [b,c], [a,d] λ is elementary if λ([i,j]) = 0   for all   [b,c] [i,j] [a,d]   except   [i,j] = [b,c], [a,c], [b,d]   or   [a,d]. Pictorially a degeneration takes PICTUREPICTURE and PICTUREPICTURE for   c=b-1, or, equivalently, PICTUREPICTURE.

Let A be the quiver (I,Ω+) with I=, Ω+ = {ii+1  |  i}. Fix an I-graded vector space V = iIVi, and let EV = ii+1 Hom(Vi,Vi+1), GLV = i GL(Vi), which acts on   EV,   and 𝒩V = {xEV  |  x  is a nilpotent element of   Hom(V,V)}. The map 𝒩V {multisegments} x λx given by λ([i]) = dim(Vi) and λ([i,j]) = rank(λ:ViVj) provides a bijection {multisegments   λ  |  λ([i])=dim(Vi)} {GLV   orbits in   𝒩V}.

Let λ and μ be multisegments and let 𝕆λ and 𝕆μ be the corresponding orbits in 𝒩V/GLV. Then the following are equivalent:

  1. λμ,
  2. 𝕆λ_𝕆μ,
  3. λ=Ri1Rirμ for some sequence of elementary degenerations Ri1,...,Rir.

If 𝕆μ𝕆λ_ then μ([i,j]) = rank(μ:ViVj) rank(λ:ViVj) = λ([i,j]).
Assume λ([i,j]) μ([i,j]) for all segments [i,j]. Find (THIS STILL NEEDS DOING) a sequence Ri1Rir of elementary degenerations which takes μ to λ, i.e. Ri1Rirμ = λ.

Hecke algebra representations

Let Hk˜ be the affine Hecke algebra at an lth root of unity so that ql=1 (all l= if desired). For each b B˜() let b = j [sj,nj) and define the   standard module M(b) = IndH˜νH˜k (s), where ν=(n1,...,nr) and k=n1++nr. The simple H˜k-modules are indexed by bB() and are determined by the equations [M(b)] = [L(b)] + b>b bB() dbb [L(b)], bB(), dbb 0, in the Grothendieck group of H˜k(q)-modules.

The Fock space representation of Uv𝔰𝔩^l

The crystal graph

Let λ = [ (λ+ρ)1 (λ+ρ)2 (λ+ρ)n (μ+ρ)1 (μ+ρ)2 (μ+ρ)n ] = ( (λ+ρ)1 (λ+ρ)2 (λ+ρ)n d1 d2 dn ] be a multisegment and assume that it is ordered so that

  1. (λ+ρ)i (λ+ρ)i+1,
  2. (μ+ρ)i (μ+ρ)i+1 if (λ+ρ)i = (λ+ρ)i+1.
These conditions are equivalent to saying that
  1. The 𝔤𝔩(n)-weight λ is integrally dominant,
  2. μ=wν where ν is integrally dominant and w is longest in its coset Wλ+ρ w Wμ+ρ.
  1. -1 above each (λ+ρ)j = i,
  2. +1 above each (λ+ρ)j = i-1,
  3. 0 above each (λ+ρ)j i,i-1.
Then, ignoring 0s, read the sequence of +1s, -1s left to right and successively cancel adjacent (-1,+1) pairs to get a sequence of the form cogood good +1 +1  +1 conormal modes -1 -1  -1 normal modes The -1s in this sequence are the normal nodes and the +1s are the conormal nodes. The good node is the leftmost normal node and the cogood node is the rightmost conormal node.

Define wt(λ) = iI -( number of boxes of content  i  in  λ )αi, and εi(λ) = (number of normal nodes), φi(λ) = ( number of conormal nodes ), e˜iλ = ( same as  λ  but with the good node   (λ+ρ)j=i   changed to  i-1 ), f˜iλ = ( same as  λ  but with the cogood node   (λ+ρ)j=i-1   changed to  i ), for each iI.

If this algorithm is being executed where I=/l then take

  1. (λ+ρ)j=l, when i=0 and (λ+ρ)j0 modl, and
  2. (λ+ρ)j=0, when i=1 and (λ+ρ)j0 modl.

  1. In type Al-1(1), B() is the connected component of in the crystal graph B˜().
  2. B() is the crystal graph of Uv-𝔤.

The crystals B(Λ)

Type Al-1: Let λ = i=1l λiϵi = iI γiωi P+, and identify λ with the partition which has λi boxes in row i. Let B(λ) = {column strict tableaux of shape   λ} and define an imbedding B(λ) B() P [ 1 1 1 2 2 2 n n n i1 i2 iλ1 iλ1+1 iλ1+λ2 ik ] where the entries i1i2ik are the entries of P read in Arabic reading order.

The tensor product representation

The l-dimensional simple Uq𝔰𝔩l-module of highest weight ω1 is given by L(ω1) = -span {v0,...,vl-1} with Uq𝔰𝔩l-action eivj = { vi-1, if   j=i, 0, if   ji, } fivj = { vi, if   j=i-1, 0, if   ji-1, } kivj = { qvi-1, if   j=i, q-1vi, if   j=i-1, vj, if   ji,i-1. } Then L(ω1)k = -span {vj1vjk  |  1j1,j2,...,jkl}. If v = vj1vjk place

  1. +1 over each vi-1 in v,
  2. -1 over each vi in v,
  3. 0 over each vj, ji,i-1.
Then the Uq𝔰𝔩l-action on L(ω1)k is given by ei(v) = v- q -(sum of  ±1s before   v/v-) v-, fi(v) = v+ q -(sum of  ±1s after   v+/v) v+, ki(v) = q (sum of  ±1s for   v) v, where the first sum is over all v- which are obtained from v by changing a vi to vi-1 and the second sum is over all v+ which are obtained from v by changing a vi-1 to vi.

The Fock space

Let μ𝔥* for 𝔤𝔩n. Define μ = -span {multisegments   λ=λ/μ}. Define an action of Uv𝔰𝔩^l on μ by eλ = c(λ/λ-)i q (sum of  ±1s before   λ/λ-) λ-, fiλ = c(λ+/λ)i q (sum of  ±1s after   λ+/λ) λ+, kiλ = q (sum of the  ±1   sequence for   λ) λ, Dλ = q( #   of boxes of content 0 in   λ ) λ.

  1. These formulas make μ into a Uv𝔰𝔩^l-module.
  2. If Lμ = [q,q-1]-span {multisegments   λ=λ/μ} so that the multisegments form a [q,q-1] basis of Lμ then e˜i [λ] = [e˜iλ]   modqLμ and f˜i [λ] = [f˜iλ]   modqLμ.

The permutations of the sequence +1,+1,...,+1,-1,-1,...,-1 are indexed by the elements of St/Sk × St-k where t is the number of nodes after (-1,+1) pairing. The group (/2)p acts on the (-1,+1) pairs by changing a pair (-1,+1) to (+1,-1). For each 1kr define uk = σSt/Sk×St-k τ(/2)t ql(σ) (-1)l(τ) (στλ[k]). Then uk = λ[k]   modqLμ and eiuk = [k]uk-1. The first statement is clear. To obtain the second statement eiuk = σSt/Sk×St-k τ(/2)t ql(σ) (-1)l(τ) (eiστλ[k]) = ei   changes a pair σSt/Sk×St-k τ(/2)t ql(σ) (-1)l(τ) (στλ[k])- + ei   changes a node σSt/Sk×St-k τ(/2)t ql(σ) (-1)l(τ) (στλ[k])- = 0+ τ(/2)t σSt/Sk×St-k ei   changes a node ql(σ) (-1)l(τ) (στλ[k])- = τ(/2)t [k] σSt/Sk×St-k ql(σ) (-1)l(τ) (στλ[k])-

A Schur-Weyl duality connection to affine Hecke algebras

A multisegment is a collection of rows of boxes (segments) placed on graph paper. We can label this multisegment by a pair of weights λ = λ1ε1 ++ λn+1 εn+1 and μ = μ1ε1 ++ μn+1 εn+1 by setting (λ+ρ)i = content of the last box in row   i, and (μ+ρ)i = (content of the last box in row   i)-1. For example 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 corresponds to λ+ρ = 77755   and μ+ρ = 22402 (MS 3) (the numbers in the boxes in the picture are the contents of the boxes). The construction forces the condition

  1. (λ+ρ)i - (μ+ρ)i 0,
    and since we want to consider unordered collections of boxes it is natural to take the following pseudo-lexicographic ordering on the segments,
  2. (λ+ρ)i (λ+ρ)i+1,
  3. (μ+ρ)i (μ+ρ)i+1 if (λ+ρ)i = (λ+ρ)i+1,
when we denote the multisegment λ/μ by a pair of weights λ,μ. In terms of weights the conditions (a), (b) and (c) can be restated as (note that in this case both λ and μ are integral)
  1. λ-μ is a weight of Vk, where k is the number of boxes in λ/μ,
  2. λ is integrally dominant,
  3. μ=wν with ν integrally dominant and w maximal length in the coset Wλ+ρ w Wν+ρ .

Let λ/μ be a multisegment with k boxes and number the boxes of λ/μ from left to right (like a book). Define H˜λ/μ =   subalgebra of   H˜k   generated by   {Xλ,Tj  |  λL,  boxj   is not at the end of its row }, so that H˜λ/μ is the "parabolic" subalgebra of H˜k corresponding to the multisegment λ/μ. Define a one-dimensional H˜λ/μ module λ/μ = vλ/μ by setting Xεi vλ/μ = q 2c(boxi) vλ/μ , and Tj vλ/μ = qvλ/μ, (MS 4) for 1ik and j such that boxj is not at the end of its row.

Let 𝔤 be of type An and let Fλ be the functor HomUh𝔤 ( M(λ), Vk ) where V=L(ω1). The standard module for the affine Hecke algebra H˜k is λ/μ = Fλ(M(μ)) (MS 5) as defined in (4.1). It follows from the above discussion that these modules are naturally indexed by multisegments λ/μ. The following proposition shows that this standard module coincides with the usual standard module for the affine Hecke algebra as considered by Zelevinsky [Ze2] (see also [Ar], [CG] and [KL]).

Let λ/μ be a multisegment determined by a pair of weights (λ,μ) with λ integrally dominant. Let λ/μ be the one dimensional representation of the parabolic subalgebra H˜λ/μ of the affine Hecke algebra H˜k defined in (???). Then λ/μ Ind H˜λ/μ H˜k (λ/μ).

To remove the constants that come from the difference between 𝔤𝔩n and 𝔰𝔩n the affine braid group action in Theorem 6.17a should be normalized so that Φk (Xε1) = q 2|μ| / (n+1) Rˇ02 and Φk(Ti) = q1(n+1) Rˇi.

By Proposition 4.3a, cMλ/μ (Vk)λ-μ as a vector space. Let {v1,v2,...,vn+1} be the standard basis of V = L(ω1) with wt(vi) = εi. If we let the symmetric group Sk act on Vk by permuting the tensor factors then (Vk)λ-μ = span-{ πv(λ-μ)  |  πSk } = span-{ πv(λ-μ)  |  πSk/Sλ-μ }, where v(λ-μ) = v1v1 λ1-μ1 vnvn λn-μn and Sλ-μ = Sλ1-μ1 ×× Sλn-μn is the parabolic subgroup of Sk which stabilizes the vector v(λ-μ) Vk. This shows that, as vector spaces, λ/μ Ind H˜λ/μ H˜k (λ/μ) = span-{ Tπvλ/μ  |  πSk/Sλ-μ } (MS 6) are isomorphic.

For notational purposes let bλ/μ = vμ+ v(λ-μ) = vμ+ vi1 vik and let b_λ/μ be the image of bλ/μ in (MVk)[λ]. Since λ is integrally dominant and b_λ/μ has weight λ it must be a highest weight vector. We will show that Xεl acts on b_λ/μ by the constant qc(boxl), where c(boxl) is the content of the lth box of the multisegment λ/μ (read left to right and top to bottom like a book).

Consider the projections prl: M(μ) Vk ( M(μ) Vl )[ λ(l) ] V(k-l) where λ(l) = μ + jl wt(vil) and pri acts as the identity on the last k-i factors of M(μ) Vk. Then b_λ/μ = prkprk-1pr1 bλ/μ, and for each 1lk, (the first l components of) prl-1 pr1 (bλ/μ) form a highest weight vector of weight λ(l) in MVl. It is the "highest" highest weight vector of ( ( M(μ) V(l-1) ) [ λ(l-1) ] V ) [ λ(l) ] (MS 7) with respect to the ordering in Lemma 4.2 and thus it is deepest in the filtration constructed there. Note that the quantum Casimir element acts on the space in (6.29) as the constant q λ(l) , λ(l) +2ρ times a unipotent transformation, and the unipotent transformation must preserve the filtration coming from Lemma 4.2. Since prl (bλ/μ) is the highest weight vector of the smallest submodule of this filtration (which is isomorphic to a Verma module by Lemma 4.2b) it is an eigenvector for the action of the quantum Casimir. Thus, by (2.11) and (2.13), Xεl acts on prl(bλ/μ) by the constant q λ(l),(l)+2ρ - λ(l-1),λ(l-1)+2ρ - ω1,ω1+2ρ = q2c(boxl), (see [LR]). Since Xεl commutes with prj for j>l this also specifies the action of Xεl on b_λ/μ = prl(bλ/μ).

The explicit R-matrix RˇVV: VVVV for this case (𝔤 of type A and V=L(ω1)) is well known (see, for example, the proof of [LR, Prop. 4.4]) and given by (vivj) q1(n+1) RˇVV = { vjvi, if   i (LESS THAN?) j, (q-q-1) vivj + vjvi, if   i (GREATER THAN?) j, qvivj, if   i=j. } Since Ti acts by RˇVV on the ith and (i+1)st tensor factors of Vk and commutes with the projection prλ it follows that Tj(b_λ/μ) = qb_λ/μ, if boxj is not a box at the end of a row of λ/μ. This analysis of the action of H˜λ/μ on b_λ/μ shows that there is an H˜k-homomorphism Ind H˜λ/μ H˜k (vλ/μ) λ/μ vλ/μ b_λ/μ. This map is surjective since λ/μ is generated by b_λ/μ (the k action on vλ-μ generates all of (Vk)λ-μ ). Finally, (6.28) guarantees that it is an isomorphism.

In the same way that each weight μ𝔥* has a normal form μ=wμ˜, with μ˜  integrally dominant, and w  maximal length in the coset   wWμ˜+ρ, every multisegment λ/μ has a normal form λ/μ = ν/(wν˜), with ν+ρ   the sequence of contents of boxes of   λ/μ, ν˜ = ν-11...1, and w   maximal length in   Wν+ρ w Wν+ρ . The element w in the normal form ν/(wν˜) of λ/μ can be constructed combinatorially by the following scheme. We number (order) the boxes of λ/μ in two different ways.

First ordering: To each box b of λ/μ associate the following triple ( content of the box to the left of   b, -(content of  b), -(row number of  b) ) where, if a box is the leftmost box in a row "the box to its left" is the rightmost box in the same row. The lexicographic ordering on these triples induces an ordering on the boxes of λ/μ.

Second ordering: To each box b of λ/μ associate the following pair ( content of  b, -(the number of box  b  in the first ordering) ). The lexicographic ordering of these pairs induces a second ordering on the boxes of λ/μ.

If v is the permutation defined by these two numberings of the boxes then w=w0vw0. For example, for the multisegment λ/μ displayed in (6.24) the numberings of the boxes are given by 21 6 10 13 18 20 5 9 12 17 19 11 16 15 1 2 4 8 14 3 7 first ordering of boxes and 3 7 12 16 19 4 8 13 17 20 11 18 21 1 2 6 9 14 5 10 15 second ordering of boxes and the normal form of λ/μ is ν = 777666555554444333321 , ν˜ = 666555444443333222210 ,   and   w=w0vw0   where v = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 15 1 21 20 14 2 6 5 4 3 19 10 9 8 7 13 12 11 18 17 16 . Let 𝔤 be of type An and V=L(ω1) and let λ/μ = Fλ (L(μ)), (MS 8) as defined in (4.1). It is known (a consequence of Proposition 6.27 and Proposition 4.3c) that λ/μ is always a simple H˜k-module or 0. Furthermore, all simple H˜k modules are obtained by this construction. See [Su] for proofs of these statements. The following theorem is a reformulation of Proposition 4.12 in terms of the combinatorics of our present setting.

Let λ/μ and ρ/τ be multisegments with k boxes (with μ and τ assumed to be integral) and let λ/μ = ν/(wν˜) and ρ/τ = γ/(vγ˜) be their normal forms. Then the multiplicities of ρ/τ in a Jantzen filtration of λ/μ are given by j0 [ (λ/μ)(j) (λ/μ)(j+1) : ρ/τ ] v 12 (l(y)-l(w)+j) = { Pwv, if   ν=γ, 0, if   νγ, } where Pwv(v) is the Kazhdan-Lusztig polynomial for the symmetric group Sk.

Theorem 6.31 says that every decomposition number for affine Hecke algebra representations is a Kazhdan-Lusztig polynomial. The following is a converse statement which says that every Kazhdan-Lusztig polynomial for the symmetric group is a decomposition number for affine Hecke algebra representations. This statement is interesting in that Polo [Po] has shown that every polynomial in 1+v0[v] is a Kazhdan-Lusztig polynomial for some choice of n and permutations v,wSn. Thus, the following proposition also shows that every polynomial arises as a generalized decomposition number for an appropriate pair of affine Hecke algebra modules.

Let λ = rr...r = (rr) and μ = 00...0 = (0r). Then, for each pair of permutations v,wSr, the Kazhdan-Lusztig polynomial Pvw(v) for the symmetric group Sr is equal to Pvw(v) = j0 [ (λ/wμ)(j) (λ/wμ)(j+1) : λ/vμ ] v 12 ( l(y)-l(w)+j ) .

Since μ+ρ and λ+ρ are both regular, Wλ+ρ = Wμ+ρ = 1 and the standard and irreducible modules λ/(wμ) and λ/(vμ) ranging over all v,wSk. Thus, this statement is a corollary of Proposition 4.12.


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[GR] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 (2004), no. 8, 1263-1283.

[Ka] J. Kamnitzer, Mirkovic-Vilonen cycles and polytopes, math.AG/050136.5.

[KM] M. Kapovich and J.J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation thoery, math.RT/0411182.

[LP] C. Lenart and A. Postnikov, Affine Weyl groups in K-theory and representation theory, math.RT/0309207.

[PR] H. Pittie and A. Ram, A Pieri-Chevalley formula in the K-theory of a G/B-bundle, Electronic Research Announcements 5 (1999), 102-107, and A Pieri-Chevalley formula for K(G/B), math.RT/0401332.

[1] A. Ram, Affine Hecke algebras and generalized standard Young tableaux, J. Algebra, 230 (2003), 367-415.

[2] A. Ram, Skew shape representations are irreducible, in Combinatorial and Geometric representation theory, S.-J. Kang and K.-H. Lee eds., Contemp. Math. 325 Amer. Math. Soc. 2003, 161-189, math.RT/0401326.

[3] RR A. Ram and J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory, in A tribute to C.S. Seshadri: Perspectives in Geometry and Representation theory, V. Lakshimibai et al eds., Hindustan Book Agency, New Delhi (2003), 428-466, math.RT/0401322.

[Sc] C. Schwer, Ph.D. Thesis, University of Wuppertal, 2006.

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