Two boundary Hecke Algebras and the combinatorics of type C

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

Last updated: 27 January 2015

Calibrated representations of Hkext

A calibrated Hkext-module is an Hkext-module M such that W0,W1,,Wk are simultaneously diagonalizable as operators on M. In the context of (2.37), M is calibrated if M=γ𝒞 Mγ,where Mγ= { mM|W0m =zmand Wim=γim fori=1,,k } (3.1) for γ=(z,γ1,,γk)𝒞. Another formulation is that M is calibrated if M has a basis of simultaneous eigenvectors for W0,,Wk. This section follows the framework of [Ram2003] in developing combinatorial tools for describing the structure and the classification of irreducible calibrated Hkext-modules. In Section 5 we will use this combinatorics to analyze and classify the Hkext-modules arise in the basic Schur-Weyl duality settings.

With notations as in the definition of 𝒲0 in (2.2), the reflection representation of 𝒲0 is the action of 𝒲0 on 𝔥=k given by w(c1,,ck)= (cw-1(1),,cw-1(k)), wherec-i=- cifori=1,2,,k. The dual space 𝔥* has basis ε1,,εk, where εi:𝔥 is the -linear map given by εi(γ1,,γk)=γi. With ε-i=-εi, the action of 𝒲0 on k produces an action on 𝔥* given by wεi=εw-1(i).

Let R+ = {ε1,,εk} {εj-εi,εj+εi|1i<jk} = {ε1,,εk} {εj-εi|1i<jk} {εj-ε-i|1i<jk} = {ε1,,εk} { εj-εi|i, j{-k,,-1,1,,k}, i<j,i-j } . If w𝒲0, the inversion set of w is R(w) = {αR+|wαR+} (3.2) = {εi|ifi>0andw(i)<0} {εj-εi|if0<i<jandw(i)>w(j)} (3.3) {εj+εi|if0<i<jand-w(i)>w(j)}. The chambers are the connected components of 𝔥\αR+𝔥α, where 𝔥α={γ𝔥|α(γ)=0}. The fundamental chamber in 𝔥 is C= {c𝔥|α(γ)>0forαR+}= {(c1,,ck)k|0<c1<c2<<ck}, and the group 𝒲0 can be identified with the set of chambers via the bijection 𝒲0 {chambers} w w-1C. Sincew-1C= { c𝔥| α(c)<0 ifαR(w)and α(c)>0 ifαR+\R(w) } , the set R(w) determines w.

Local regions

For γ=(γ1,,γk)(×)k define Z(γ) = {εi|γi=±1} {εj-εi|0<i<j,γiγj-1=1} {εj+εi|0<i<j,γiγj=1}, P(γ) = {εi|γi{(t012tk12)±1,(-t0-12tk12)±1}} {εj-εi|0<i<j,γiγj-1=t±1} {εj+εi|0<i<j,γiγj=t±1}. (3.4) Using the conversion from γi to ci as in (2.32), let γi=-tci, and-tr1=- tk12t0-12 and-tr2= tk12t012, (3.5) so that -t±r1 and -t±r2 are the eigenvalues of W1 that cause τ02 to have a nonzero kernel (see (2.42)). Then, for c=(c1,,ck)k let c-i=-ci and define Z(c) = {εi|ci=0} {εj-εi|0<i<jandcj-ci=0} {εj+εi|0<i<jandcj+ci=0} (3.6) P(c) = {εi|ci{±r1,±r2}} {εj-εi|0<i<jandcj-ci=±1} (3.7) {εj+εi|0<i<jandcj+ci=±1}. A local region is a pair (c,J) with ck and JP(c). The set of standard tableaux of shape (c,J) is (c,J)= { w𝒲0| R(w)Z(c) =,R(w) P(c)=J } . (3.8)

As in [Ram2003, §5 and §8] the local regions (c,J) and standard tableaux w(c,J) can be converted to configurations of boxes κ and standard tableaux S of shape κ similar to those that are familiar in the literature on irreducible representations of Weyl groups of classical types. As explained in [Ram2003, §5.11], the definitions of Z(c) and P(c) make it possible to view the general case ck as pieced together from the cases c(+β)k where β runs over a set of representatives of the -cosets in . Below we make the conversion between local regions and configurations of boxes explicit for the cases when ck and c(+12)k. These are the cases that appear in the Schur-Weyl duality approach to the representations of Hkext which we explore in Section 5. As in [Ram2003, §8], it is also true that these cases are sufficient to determine the general c(+β)k setting but we shall not do this in detail here.

Let (c,J) be a local region with c=(c1,,ck), ckor c(+12)k ,and0c1 ck. (3.9) Start with an infinite arrangement of NW to SE diagonals, numbered consecutively from or +12, increasing southwest to northeast (see Example 1). The configuration κ of boxes corresponding to the local region (c,J) has 2k boxes (labeled box-k,,box-1,box1,,boxk) with the following conditions.

(κ1) Location: boxi is on diagonal ci, where c-i=-ci for i{-k,,-1}.
(κ2) Same diagonals: boxi is NW of boxj if i<j and boxi and boxj are on the same diagonal.
(κ3) Adjacent diagonals:
If εj-εiJ, then boxj is NW (strictly north and weakly west) of boxi: j i
If εj-εiP(c)-J, then boxj is SE (weakly south and strictly east) of boxi: i j
(κ4) Markings: There is a marking on each of the diagonals r1, -r1, r2 and -r2.
If εiJ, boxi is NW of the marking in diagonal ci: i
If εiP(c)-J, then boxi is SE of the marking in diagonal ci: i
Condition (κ1) enables the values (c-k,,c-1,c1,,ck) to be read off of configuration κ. The sets Z(c), P(c), and J can also be determined from the configuration κ since Z(c) = {εi|0<iandboxiis in diagonal 0} { εj-εi| 0<i<jandboxi andboxjare in the same diagonal } { εj+εi| 0<i<jandboxi andboxjare both in diagonal 0 } , P(c) = {εi|0<iandboxiis in diagonalr1orr2}, { εj-εi| 0<i<jandboxi andboxjare in adjacent diagonals } { εj+εi| 0<i<jandbox-i andboxjare in adjacent diagonals } , and J = {εiP(c)|boxiis NW of the marking} { εj-εiP(c) |boxjis northwest of boxi } { εj+εiP(c) |boxjis northwest of box-i } . A standard filling of the boxes of κ is a bijective function S:κ{-k,,-1,1,,k} such that
(S1) Symmetry: S(box-i)=-S(boxi).
(S2) Same diagonals:
If 0<i<j and boxi and boxj are on the same diagonal then S(boxi)<S(boxj).
(S3) Adjacent diagonals:
If 0<i<j, boxi and boxj are on adjacent diagonals, and boxj is NW of boxi, then S(boxj)<S(boxi).
If 0<i<j, boxi and boxj are on adjacent diagonals, and boxj is SE of boxi, then S(boxj)>S(boxi).
(S4) Markings:
If boxi is on a marked diagonal and is SE of the marking, then S(boxi)>0.
If boxi is on a marked diagonal and is NW of the marking, then S(boxi)<0.
The identity filling of a configuration κ is the filling F of the boxes of κ given by F(boxi)=i, for i=-k,,-1,1,,k. The identity filling of κ is usually not a standard filling of κ (see Example 1).

Let k=4, r1=1, and r2=3. Consider c=(-3,-2,-2,2,2,3). Then Z(c)={ε1-ε2} and P(c)= {ε3,ε1-ε3,ε2-ε3}. The box configurations corresponding to J={ε2-ε3} and J={ε3,ε1-ε3,ε2-ε3} (filled with their identity fillings) are 1 3 2 -2 -3 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5 3 1 2 -2 -1 -3 0 1 2 3 4 5 -1 -2 -3 -4 -5 J={ε2-ε3} J={ε3,ε1-ε3,ε2-ε3} For both configurations, the identity filling is not a standard filling. Examples of standard fillings of the configuration corresponding to J={ε2-ε3} include 1 2 3 -3 -2 -1 , , -1 2 3 -3 -2 -1 ,and -2 1 3 -3 -1 -2 ,but not -3 -2 1 -1 2 3 .

The proof of the following proposition is a straightforward, though slightly tedious, check that the conditions R(w)Z(c)= and R(w)P(c)=J from (3.8) convert to the conditions (S2), (S3), (S4) on standard fillings of shape κ. The proof is similar to the proof of [Ram2003, Thm. 5.9].

Let κ be a configuration of boxes corresponding to a local region (c,J) with ck or c(+12)k. For w𝒲0 let Sw be the filling of the boxes of κ given by Sw(boxi)=w (i),for i=-k,,-1,1, k. The map (c,J) {standard fillingsSof the boxes ofκ} w Sw is a bijection.

Let k=12, r1=32, r2=152, c=(12,12,32,32,52,92,112,132,132,152,152,172) and J= { ε3,ε10, ε3-ε2, ε4-ε2, ε5-ε4, ε8-ε7, ε10-ε8, ε10-ε9, ε11-ε9, ε12-ε10, ε12-ε11 } Let w= ( 1 2 3 4 5 6 7 8 9 10 11 12 -9 10 -8 7 6 3 4 1 5 -11 2 -12 ) (c,J). Then, for the corresponding configuration of boxes κ, the identity filling F, and the standard filling Sw corresponding to w are F= 12 10 8 11 6 7 9 -2 1 3 5 -4 4 -5 -3 -1 2 -9 -7 -6 -11 -8 -10 -12 172 152 12 -12 -152 -172 andSw= -12 -11 1 2 3 4 5 -10 -9 -8 6 -7 7 -6 8 9 10 -5 -4 -3 -2 -1 11 12 172 152 12 -12 -152 -172

Borrowing a physical intuition, configurations are invariant under sliding boxes along diagonals like beads on an abacus, so long as boxes that run into each other are not allowed to exchange places, i.e. for most c, c+1 c = c+1 c c+1 c . Then by arranging configurations so that the boxes are packed together, standard fillings of configurations are exactly analogous to standard tableaux for partitions.

The only exception to this physical intuition is for boxes on the diagonals ±12. Note that if ci=12, then boxi and box-i are on adjacent diagonals. However, since 2εi=εi-ε-iR+ and therefore never in P(c), the relative positions of boxi and box-i will never be recorded in the set J. For example, in Figure 2, the point where (c1,c2)=(12,12) has two configurations, each with two boxes overlapping in indication that boxi and box-i may "slide past each other". The drawing (with boxes filled in the identity filling) represents the equivalence of -12 12 -2 -1 1 2 and -12 12 -2 1 -1 2 , where box1 and box-1 can move freely past each other, and represents the equivalence of 12 -12 2 1 -1 -2 and 12 -12 2 -1 1 -2 , where box2 and box-2 can move freely past each other. In these two examples ε1-ε-2P(c) and ε2-ε-1P(c) and so the relative orientation of box2 and box-1 and the relative orientation of box1 and box-2 are recorded in J. Each configuration has exactly two standard fillings.

Classifying and constructing calibrated representations

Theorem 3.3 below provides an indexing of the calibrated irreducible Hkext-modules by skew local regions. A skew local region is a local region (c,J), c=(c1,,ck), such that if w(c,J) then wc=((wc)1,,(wc)n) satisfies (wc)10, (wc)20, (wc)1 -(wc)2, (wc)i (wc)i+1 fori=1,,k-1, and(wc)i (wc)i+2 fori=1,,k-2. (3.10) Theorem 3.3 is completely analogous to the same theorem for the case t12=t012=tk12 in [Ram2003, Theorem 3.5]. As explained in the discussion and remarks before [Ram2003, Lemma 3.1] in [Ram2003, §3], this depends on getting exactly the right definition of skew local region, which is accomplished by a detailed computation of the irreducible representations in rank two cases. More specifically, for I{0,,k}, let HI be the subalgebra of Hkext generated by {Ti}iI and [W1±1,,Wk±1]. Then the conditions in (3.10) guarantee that for w(c,J) and i,j{0,1,,k-1}, there exists a calibratedH{i,j} -moduleMwith Mwcgen0. For the cases where H{i,j} is of type A×A1 or of type A2 this is checked in [Ram2002]. However, when H{i,j} is of type C2 and there are three distinct parameters we do not know a reference for this and so, in the effort to provide a more complete presentation, we have done the appropriate analysis in Section 4 for all generic choices of the three parameters t12, t012, and tk12, as defined in (??), which we rewrite here: t12 is not a root of unity and t012tk12,- t0-12tk12 {1,-1,t±12,-t±12,t±1,-t±1} andt012 tk12 (-t0-12tk12)±1.

Assume t12, t012, and tk12 are invertible, t12 is not a root of unity, and t012tk12,- t0-12tk12 {1,-1,t±12,-t±12,t±1,-t±1} andt012 tk12 (-t0-12tk12)±1.

(a) Let (c,J), c=(c1,,ck), be a skew local region and let z×. Define Hk(z,c,J)= span{vw|w(c,J)} (3.11) so that the symbols vw are a labeled basis of the vector space Hk(z,c,J). Let γi=-tcifor i=1,2,,k,and γ0=z γw-1(1)-1 γw-1(k)-1. Then the following formulas make Hk(z,c,J) into an irreducible Hkext-module: PW1Wkvw=zvw, Pvw=γ0 vw,Wivw =γw-1(i) vw, (3.12) Tivw = [Ti]wwvw+ -([Ti]ww-t12)([Ti]ww+t-12) vsiw,fori =1,,k-1, (3.13) T0vw = [T0]wwvw+ -([T0]ww-t012)([T0]ww+t0-12) vs0w, (3.14) where vsiw=0 if siw(c,J) and [Ti]ww= t12-t-12 1-γw-1(i)γw-1(i+1)-1 ,and [T0]ww= (t012-t0-12)+ (tk12-tk-12) γw-1(1)-1 1-γw-1(1)-2 . (3.15)
(b) The map ××{skew local regions(c,J)} {irreducible calibratedHkext-modules} (z,c,J) Hk(z,c,J) is a bijection.


This result follows from [Ram2003, Theorems 3.2 and 3.5]. It is only necessary to establish that the formulas in (3.12), (3.13) and (3.14) are correct. These are derived in a similar manner to [Ram2003, Proposition 3.3] as follows. As in [Ram2003, Theorem 3.2], if M is an irreducible calibrated Hkext-module then M=w𝒲0 Mwγgen, withdim(Mwγgen) =1ifMwγgen0. For w𝒲0, if Mwγgen0, let vw be a nonzero vector in Mwγgen; otherwise if Mwγgen=0, let vw=0. By (2.38), τivw=[Ti]siw,wvsiw for some constant [Ti]siw,w and the definition of τi in (2.35) gives that Tivw= t12-t-12 1-γw-1(i)γw-1(i+1)-1 vw+[Ti]siw,w vsiw,fori= 1,,k, (3.16) and T0vγ= (t012-t0-12)+ (tk12-tk-12) γw-1(1)-1 1-γw-1(1)-2 vw+[T0]s0w,w vs0w. (3.17) Thus T0 is an operator on the subspace span{vw,vs0w} satisfying (T0-t012)(T0+t-12)=0 by (H). Restricting to the action on span{vw,vs0w}, the formulas in (3.14) now follow from the following argument about general 2×2 matrices.

If a 2×2 matrix [T0] has eigenvalues α1 and α2, [T0]= ( [T0]ww [T0]w,s0w [T0]s0w,w [T0]s0w,s0w ) ,then ([T0]-α1) ([T0]-α2)=0 is the characteristic polynomial for [T0], and it follows that Tr([T0]) = [T0]ww+ [T0]s0w,s0w= α1+α2,and det([T0]) = [T0]ww [T0]s0w,s0w- [T0]w,s0w [T0]s0w,w= α1α2. Thus -[T0]w,s0w [T0]s0w,w = α1α2- [T0]ww [T0]s0w,s02= α1α2- [T0]ww ((α1+α2)-[T0]ww) = α1α2- (α1+α2) [T0]ww+ ([T0]ww)2= ([T0]ww-α1) ([T0]ww-α2). Choosing a normalization of vs0w so that the matrix of [T0] is symmetric, we have [T0]w,s0w=[T0]s0w,w and [T0]s0w,w= ([T0]s0w,w)2= [T0]w,s0w [T0]w,s0w = - ([T0]ww-α1) ([T0]ww-α2) .

Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type C Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).

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