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A preorder-free construction of the Kazhdan-Lusztig representations of Hecke algebras Hn(q) of symmetric groups

Charles Buehrle and Mark Skandera

Department of Mathematics Lehigh University

August 2010

The Hecke algebra, Hn(q)

C[q

1 2 , q

¯1

2 ]-algebra generated by {

Tsi | 1 ≤ i ≤ n − 1} with relations

• T 2

si = (q

1 2 − q

¯1

2 )

Tsi + Te, 1 ≤ i ≤ n − 1

• Tsi

Tsj Tsi = Tsj Tsi Tsj, |i − j| = 1

• Tsi

Tsj = Tsj Tsi, |i − j| ≥ 2. The natural basis of Hn(q) is the set { Tv = Tsi1 · · · Tsiℓ(v) | v ∈ Sn}. Notice that Hn(1) ∼ = C[Sn]. For a v ∈ Sn let P(v), Q(v) be the tableaux obtained from Robinson-Schensted column insertion.

The Kazhdan-Lusztig basis of Hn(q)

In [Kazhdan and Lusztig, 1979] a certain basis of Hn(q) is defined for each v ∈ Sn to be C ′

v =

• u≤v

(q

1 2 )ℓ(v)−ℓ(u)Pu,v(q)

Tu, where Pu,v(q) are the Kazhdan-Lusztig polynomials. Although, Pu,v(q) ∈ N[q] there is no simple combinatorial description of the coefficients.

Kazhdan-Lusztig preorders on Hn(q)

Kazhdan-Lusztig preorders help to construct representations.

Right preorder

◮ v ⋖R u if av = 0 in C ′ u

Tw =

z∈Sn azC ′ z, for some w. ◮ The right preorder ≤R is the transitive closure of ⋖R.

Example

For S3 the Hasse diagram of the right preorder is 123 213

• 312
• 231
• 132
• 321

Kazhdan-Lusztig representations of Hn(q)

For λ ⊢ n choose a standard λ-tableau, T, and v such that Q(v) = T. Define K λ = span{C ′

u | Q(u) = T}

=

def

span{C ′

u | u ≤R v}/span{C ′ u | u <R v},

where u <R v means u ≤R v ≤R u. Matrix representations of Hn(q) are obtained by right multiplication of Tsi on the “basis”. X λ

K : Hn(q) → End((C[q

1 2 , q

¯1

2 ])d)

Example

X (2,1)

K

( Ts1) =

• −q

¯1

2

1 q

1 2

• X (2,1)

K

( Ts2) =

• q

1 2

1 −q

¯1

2

Kazhdan-Lusztig representation of H3(q)

Choose λ = (2, 1) and tableau T = 1 2 3 . So we have that K λ = span{C ′

213, C ′ 312}.

C ′

213

Ts1 = −q

¯1

2 C ′

213

C ′

312

Ts1 = q

1 2 C ′

312 + C ′ 321 + C ′ 213.

C ′

321 is not in our spanning set!

But, 321 <R 312. So we can ignore C ′

321 due to the quotient.

Thus X λ

K(

Ts1) =

• −q

¯1

2

1 q

1 2

• .

Quantum polynomial ring

Define A(n; q) = C[q

1 2 , q

¯1

2 ] x1,1, . . . , xn,n, modulo

xi,ℓxj,k = xj,kxi,ℓ, xi,ℓxi,k = q

1 2 xi,kxi,ℓ,

xj,kxi,k = q

1 2 xi,kxj,k,

xj,ℓxi,k = xi,kxj,ℓ + (q

1 2 − q

¯1

2 )xi,ℓxj,k,

for 1 ≤ i < j ≤ n,1 ≤ k < ℓ ≤ n. The relations can be remembered using the 2 × 2 submatrix xi,k xi,ℓ xj,k xj,ℓ

The immanant space and Kazhdan-Lusztig immanants

Convenient monomial notation: xv,w = xv1,w1 · · · xvn,wn.

The immanant space

span{xe,v | v ∈ Sn} an n! dimensional subspace of A(n; q). In [Du, 1992] a dual canonical basis called Kazhdan-Lusztig immanants was defined for each u ∈ Sn Immu(x) =

• v≥u

(−q

1 2 )ℓ(u)−ℓ(v)Pw0u,w0v(q)xe,v,

where Pw0u,w0v(q) are the inverse Kazhdan-Lusztig polynomials.

Generalized submatrices

For n-element multisets of [n] L = (ℓ(1), . . . , ℓ(n)) and M = (m(1), . . . , m(n)) define xL,M =    xℓ(1),m(1) · · · xℓ(1),m(n) . . . ... . . . xℓ(n),m(1) · · · xℓ(n),m(n)    .

Example

L = (1, 1, 2) and M = (2, 3, 3) xL,M =   x1,2 x1,3 x1,3 x1,2 x1,3 x1,3 x2,2 x2,3 x2,3   .

Kazhdan-Lusztig representations of Hn(q), again

For λ ⊢ n choose a standard λ-tableau, T, and v such that Q(v) = T. Define V λ = span{Immu(x) | Q(u) = T} =

def

span{Immu(x)|u ≥R v}/span{Immu(x)|u >R v}. Hn(q) acts on V λ by Tu acting on the monomial basis {xe,v | v ∈ Sn}. X λ

V : Hn(q) → End((C[q

1 2 , q

¯1

2 ])d)

Theorem

For any h ∈ Hn(q), X λ

V (h) = X λ K(h).

Kazhdan-Lusztig representation of H3(q), again

Choose λ = (2, 1) and tableau T = 1 2 3 . So we have that V λ = span{Imm312(x), Imm213(x)}. Imm312(x) Ts1 = −q

¯1

2 Imm312(x)

Imm213(x) Ts1 = q

1 2 Imm213(x) + Imm123(x) + Imm312(x).

Imm123(x) is not in our spanning set! 213 <R 123. So we can ignore Imm123(x) due to the quotient. Thus X λ

V (

Ts1) =

• −q

¯1

2

1 q

1 2

• .

Vanishing of Kazhdan-Lusztig immanants

Let L an n-element multiset of [n].

Theorem

If ℓ(i) = ℓ(i + 1) in L and siu > u then Immu(xL,[n]) = 0. For n × n matrix A µ(A) = row multiplicity partition of A. Dominance order of partitions, λ µ if k

i=1 λi ≤ k i=1 µi, for all k.

Theorem

If sh(u) µ(xL,[n]) then Immu(xL,[n]) = 0. These results are quantum analogues to results in [Rhoades and Skandera, 2009].

Quotient-free Kazhdan-Lusztig representations of Hn(q)

For λ ⊢ n, define the multiset L = (1λ1, . . . , nλn). Define W λ = span{Immu(xL,[n]) | Q(u) = T(λ)}. Matrix representations obtained by the action of Hn(q) on basis of W λ. X λ

W : Hn(q) → End((C[q

1 2 , q

¯1

2 ])d)

Theorem

For any h ∈ Hn(q), X λ

W (h) = X λ V (h) = X λ K(h).

This result is the Hn(q) analog of a result in [B. and Skandera, 2010].

Quotient-free Kazhdan-Lusztig representation of H3(q)

Choose λ = (2, 1). We have that W λ = span{Imm312(x112,123), Imm213(x112,123)}. Imm312(x112,123) Ts1 = −q

¯1

2 Imm312(x112,123)

Imm213(x112,123) Ts1 = q

1 2 Imm213(x112,123) + Imm312(x112,123)

+Imm123(x112,123). sh(123) = (1, 1, 1) ≺ µ(x112,123) = (2, 1). So Imm123(x112,123) = 0. Thus X λ

W (

Ts1) =

• −q

¯1

2

1 q

1 2

• .

Thank You

• B. and S.

Relations between the Clausen and Kazhdan-Lusztig representations

• f the symmetric group.

To appear in J. Pure Appl. Algebra, 2010. Michael Clausen Multivariate polynomials, standard tableaux, and representations of symmetric groups

• J. Symbolic Comput., 11:483–522, 1991.
• J. Du

Canonical bases for irreducible representations of quantum GLn.

• Bull. London Math. Soc., 24(4):325-334, 1992.
• D. Kazhdan and G. Lusztig.

Representations of Coxeter groups and Hecke algebras.

• Invent. Math., 53:165–184, 1979.
• B. Rhoades and S.

Bitableaux and the dual canonical basis of the polynomial ring. To appear in Adv. in Math., 2009.