Hecke Algebras Of Type Research Articles

Quantum Schur superalgebras and Kazhdan–Lusztig combinatorics

We introduce the notion of quantum Schur (or q -Schur) superalgebras. These algebras share certain nice properties with q -Schur algebras such as the base change property, the existence of canonical Z [ v , v − 1 ] -bases, the duality relation with Manin’s quantum matrix superalgebra A ( m | n ) , and the bridging role between quantum enveloping superalgebras of g l ( m | n ) and the Hecke algebras of type A . We also construct a cellular Q ( υ ) -basis and determine its associated cells, called supercells, in terms of a Robinson–Schensted–Knuth supercorrespondence. In this way, we classify all irreducible representations over Q ( υ ) via supercell modules.

Quantum symmetric pairs and representations of double affine Hecke algebras of type $${C^\vee C_n}$$

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.

The Ariki–Terasoma–Yamada tensor space and the blob algebra

We show that the Ariki–Terasoma–Yamada tensor module and its permutation submodules M ( λ ) are modules for the blob algebra when the Ariki–Koike algebra is a Hecke algebra of type B. We show that M ( λ ) and the standard modules Δ ( λ ) have the same dimensions, the same localization and similar restriction properties and are equal in the Grothendieck group. Still we find that the universal property for Δ ( λ ) fails for M ( λ ) , making M ( λ ) and Δ ( λ ) different modules in general. Finally, we prove that M ( λ ) is isomorphic to the dual Specht module for the Ariki–Koike algebra.

A Specht filtration of an induced Specht module

Let H n be a (degenerate or non-degenerate) Hecke algebra of type G ( ℓ , 1 , n ) , defined over a commutative ring R with one, and let S ( μ ) be a Specht module for H n . This paper shows that the induced Specht module S ( μ ) ⊗ H n H n + 1 has an explicit Specht filtration.

Induced Discrete Series Representations for Hecke Algebras of Types and

Recently, Delorme and Opdam [7] have generalized the theory of R-groups towards affine Hecke algebras with unequal labels. We apply their results in the case where the affine Hecke algebra is of type or , and the induced discrete series representation has positive central character. We show that the R-group of such an induced representation is isomorphic to , and that the representation decomposes multiplicity-free into irreducible summands. The number d is calculated by combinatorial means.

Lefschetz Property, Schur–Weyl Duality and a q-Deformation of Specht polynomial

We describe the Schur–Weyl duality for a polynomial representation of the quantum group and the Hecke algebra of type A from a viewpoint of a q-analog of the strong Lefschetz property. A q-deformation of the Specht polynomial appears as a constituent of bases for irreducible components.

The Mullineux Map for RoCk Blocks

We study the effect of tensoring simple modules in RoCK blocks (also known as Rouquier blocks) of symmetric groups with the one-dimensional sign representation, and the analogue for Hecke algebras of type A. We find an explicit and very neat description. We also describe p-regularization for partitions in these blocks.

Generalized Graded Hecke Algebras of Types B and D

For a root system of type B we study an algebra similar to a graded Hecke algebra, isomorphic to a subalgebra of the rational Cherednik algebra. We introduce principal series modules over it and prove an irreducibility criterion for these modules. We deduce similar results for an algebra associated to a root system of type D.

Hecke algebras of classical type and their representation type

The purpose of this article is to determine the representation type for all of the Hecke algebras of classical type. To do this, we combine methods from our previous work, which is used to obtain information on their Gabriel quivers, and recent advances in the theory of finite-dimensional algebras. The principal computation is for Hecke algebras of type B with two parameters. Then, we show that the representation type of Hecke algebras is governed by their Poincare polynomials.

Schur–Weyl reciprocity between quantum groups and Hecke algebras of type [formula omitted

Let U q ( gl m ⊕ p ) be the quantized universal enveloping algebra of gl m ⊕ p . Let θ be the automorphism of U q ( gl m ⊕ p ) which is defined on generators by E i ↦ E i − m , F i ↦ F i − m , K j ↦ K j − m for any i ∈ Z / p m Z ∖ { 0 ¯ , m ¯ , … , ( p − 1 ) m ¯ } and any j ∈ Z / p m Z . Let H ( p , p , n ) be the Hecke algebra of type G ( p , p , n ) with parameters q , ε , where ε is a primitive pth root of unity. In this paper we establish a Schur–Weyl reciprocity between H ( p , p , n ) and a twisted tensor product of U q ( gl m ⊕ p ) and the group algebra for 〈 θ 〉 (a cyclic group of order p) by using the results in [J. Hu, J. Algebra 274 (2004) 446–490].

Standard Modules for Tabular Algebras

We introduce cell modules for the tabular algebras defined in a previous work; these modules are analogous to the representations arising from left Kazhdan–Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type A equipped with its Kazhdan–Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.

The computational complexity of rules for the character table of Sn

The Murnaghan–Nakayama rule is the classical formula for computing the character table of S n . Y. Roichman (Adv. Math. 129 (1997) 25) has recently discovered a rule for the Kazhdan–Lusztig characters of q Hecke algebras of type A, which can also be used for the character table of S n . For each of the two rules, we give an algorithm for computing entries in the character table of S n . We then analyze the computational complexity of the two algorithms, and in the case of characters indexed by partitions in the ( k,ℓ) hook, compare their complexities to each other. It turns out that the algorithm based on the Murnaghan–Nakayama rule requires far less operations than the other algorithm. We note the algorithms’ complexities’ relation to two enumeration problems of Young diagrams and Young tableaux.

Modular representations of Hecke algebras of type G( p, p, n)

Hecke algebras for the complex reflection groups G( r, p, n) (where p⩾1 and p∣ r) were introduced in the work of Ariki [J. Algebra 177 (1995) 164–185], Broué and Malle [Astérisque 212 (1993) 119–189]. In this paper we consider modular representation theory for these algebras in the case where r= p. We assume that the field K contains a primitive pth root of unity ε. Our method is to study the restrictions of Specht modules S λ for Hecke algebras H p,n of type G( p,1, n) with parameters ( q;1, ε,…, ε p−1 ). Suppose that f( q, ε)≠0 in K (see 4.7 for definition of f( q, ε)). For any multipartition λ=( λ (1),…, λ ( p) ) of n, we prove that S λ↓ H p,p,n ≅ S λ[k]↓ H p,p,n for any 1⩽ k⩽ p−1, where λ[ k]=( λ ( k+1) , λ ( k+2) ,…, λ ( p) , λ (1), λ (2),…, λ ( k) ); and if k is the smallest positive integer such that λ= λ[ k] (hence k∣ p), we explicitly decompose S λ↓ H p,p,n into a direct sum of p/ k smaller H p,p,n -submodules with the same dimensions. As a result, when f( q, ε)≠0 in K, we show that H p,p,n is split over K and get a complete classification of all the absolutely irreducible H p,p,n -modules. This generalizes earlier work of [C. Pallikaros, J. Algebra 169 (1994) 20–48] and [J. Hu, Manuscripta Math. 108 (2002) 409–430] on Hecke algebras of type D n (which are included as a special case of our main results).

On double centralizer properties between quantum groups and Ariki–Koike algebras

In this paper, the double centralizer properties for tensor spaces as bimodules over quantum groups and Ariki–Koike algebras as well as their relations with Shoji's algebra H h (see [J. Algebra 226 (2000) 818–856]) are studied. We proved that, under the separation condition (see [J. Reine Angew. Math. 513 (1999) 53–69]), the natural homomorphism from the Ariki–Koike algebra to the endomorphism algebra of tensor space as module over quantum group is surjective. We also show that the natural homomorphism from Shoji's algebra H h to that endomorphism algebra is always surjective, and H h is actually isomorphic to a direct sum of some matrix algebras over some Hecke algebras of type A.

From double affine Hecke algebras to quantized affine Schur algebras

We prove an equivalence between some category of modules of the double affine Hecke algebra of type A and another of the quantized affine Schur algebra.

Crystal bases and simple modules for Hecke algebra of type Dn

Let H C (B n) (respectively H C (D n) ) be the Hecke algebra of type B n (respectively of type D n ) over the complex numbers field C . Let ζ be a primitive 2ℓth root of unity in C . For any Kleshchev bipartition (with respect to ( ζ,1,−1)) λ=( λ (1), λ (2)) of n, let D λ be the corresponding irreducible H C (B n) -module. In the present paper we explicitly determine which D λ split and which D λ remains irreducible when restricts to H C (D n) . This yields a complete classification of all the simple modules for Hecke algebra H C (D n) . Our proof makes use of the crystal bases theory for the Fock representation of the quantum affine algebra U q( sl 2ℓ) and deep result of Ariki's proof of LLT's conjecture [J. Math. Kyoto Univ. 36 (1996) 789–808].

The distinguished involutions with a-value n2−3 n+3 in the Weyl group of type Dn

Let ( W, S) be a Weyl group and H its associated Hecke algebra. Let A= Z[u,u −1] be the Laurent polynomial ring. Kazhdan and Lusztig [Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) 165–184] introduced two A -bases { T w } w∈ W and { C w } w∈ W for the Hecke algebra H associated to W. Let Y w =∑ y⩽ w u l( w)− l( y) T y . Then { Y w } w∈ W is also an A -base for the Hecke algebra. In this paper we give an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of Y x 's in the Hecke algebra of type D n . In fact, this gives also an explicit expression for certain Kazhdan–Lusztig basis elements C w as A -linear combination of T x 's in the Hecke algebra of type D n . Thus we describe also explicitly the Kazhdan–Lusztig polynomials for certain elements of the Weyl group. We study also the joint relation among some elements in W and some distinguished involutions with a-value n 2−3 n+3 in the Weyl group of type D n .

Expression of certain Kazhdan–Lusztig basis elements Cw over the Hecke algebra of type An

In this paper we determine expression of Kazhdan–Lusztig basis elements C w over the Hecke Algebra of Type A n where w = a ( j , i ) is an element in Weyl group of type A n . We also find joined relation of some elements in W .

Schur-Weyl reciprocity between quantum groups and Hecke algebras of type $G(r,1,n)$

Schur-Weyl reciprocity between quantum groups and Hecke algebras of type $G(r,1,n)$

The Hecke algebras of type B and D and subfactors

The Hecke algebras of type B and D and subfactors