The induction coefficients of the Hecke algebra and the Clebsch–Gordan coefficients of the quantum group SUq(N). II. General Gel’fand basis

1993 ◽  
Vol 34 (9) ◽  
pp. 4316-4329 ◽  
Author(s):  
Feng Pan ◽  
Jin‐Quan Chen
Keyword(s):  
Quantum Group ◽  
Hecke Algebra

scholarly journals Set-theoretic Yang–Baxter & reflection equations and quantum group symmetries

2021 ◽  
Vol 111 (4) ◽  
Author(s):  
Anastasia Doikou ◽  
Agata Smoktunowicz
Keyword(s):  
Quantum Group ◽  
Transfer Matrix ◽  
Integrable Systems ◽  
Hecke Algebra ◽  
Special Choice ◽  
Quantum Integrable Systems ◽  
Double Row ◽  
Reflection Equations ◽  
Group Symmetries ◽  
Reflection Algebra

AbstractConnections between set-theoretic Yang–Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra $${\mathcal {H}}_N(q=1)$$ H N ( q = 1 ) . We show in the case of the reflection algebra that there exists a “boundary” finite sub-algebra for some special choice of “boundary” elements of the B-type Hecke algebra $${\mathcal {B}}_N(q=1, Q)$$ B N ( q = 1 , Q ) . We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings and also allow the investigation of the symmetries of the double row transfer matrix.

CHARACTER TABLE OF HECKE ALGEBRA OF TYPE AN-1 AND REPRESENTATIONS OF THE QUANTUM GROUP Uq(gln+1)

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 977-984 ◽  
Author(s):  
Kimio UENO ◽  
Youichi SHIBUKAWA
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Hecke Algebra ◽  
Type A ◽  
Character Table

A q-analogue of the Frobenius formula is proved by means of the quantum groups Uq(gln+1), Aq(GLn+1) and Iwahori's Hecke algebra of type AN-1, and then, the character table of this Hecke algebra is investigated.

scholarly journals Commuting Families in Hecke and Temperley-Lieb Algebras

2009 ◽  
Vol 195 ◽  
pp. 125-152 ◽  
Author(s):  
Tom Halverson ◽  
Manuela Mazzocco ◽  
Arun Ram
Keyword(s):  
Quantum Group ◽  
Hecke Algebra ◽  
Type A ◽  
Irreducible Representations ◽  
Casimir Element ◽  
Affine Hecke Algebra

AbstractWe define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

scholarly journals Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl
Keyword(s):  
Preceding Paper ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Special Points ◽  
Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

scholarly journals The Real Forms of the Fractional Supergroup SL(2,C)

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 933
Author(s):  
Yasemen Ucan ◽  
Resat Kosker
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
The Real ◽  
And Mathematics ◽  
Real Forms

The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group SL2,C is one of these important groups. There are real forms of the classical Lie group SL2,C and the quantum group SL2,C in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with A3NSL2,C, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups A31SL2,C and A32SL2,C.

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