scholarly journals Weights of Markov traces on Hecke algebras

1999 ◽  
Vol 1999 (508) ◽  
pp. 157-178 ◽  
Author(s):  
R. C Orellana
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Markov Trace

Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

scholarly journals Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

2018 ◽  
Vol 2020 (1) ◽  
pp. 214-286 ◽  
Author(s):  
Maria Chlouveraki ◽  
Jesús Juyumaya ◽  
Konstantinos Karvounis ◽  
Sofia Lambropoulou
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Type A ◽  
Closed Formula ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Oriented Link ◽  
Linking Numbers ◽  
Markov Trace ◽  
Knots And Links

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

scholarly journals AN INVARIANT FOR SINGULAR KNOTS

2009 ◽  
Vol 18 (06) ◽  
pp. 825-840 ◽  
Author(s):  
J. JUYUMAYA ◽  
S. LAMBROPOULOU
Keyword(s):  
Hecke Algebras ◽  
Quadratic Relation ◽  
Link Invariant ◽  
Topological Interpretation ◽  
Type Invariant ◽  
Singular Link ◽  
Markov Trace ◽  
Skein Relation ◽  
Singular Knots

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).

scholarly journals A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

2018 ◽  
Vol 21 (03) ◽  
pp. 1850015
Author(s):  
Takehiro Hasegawa ◽  
Hayato Saigo ◽  
Seiken Saito ◽  
Shingo Sugiyama
Keyword(s):  
Inversion Formula ◽  
Probabilistic Approach ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Quantum Probability ◽  
Probabilistic Interpretation ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
The Subject

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

scholarly journals Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)

1998 ◽  
Vol 50 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Tom Halverson ◽  
Arun Ram
Keyword(s):  
Symmetric Group ◽  
Infinite Series ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Diagonal Matrix ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Irreducible Characters ◽  
Complex Reflection ◽  
Complex Reflection Groups

AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

Integral Hecke Algebras for Finite Generalized Polygons

2011 ◽  
Vol 18 (02) ◽  
pp. 259-272
Author(s):  
İlhan Hacıoglu
Keyword(s):  
Normal Form ◽  
Incidence Matrix ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Generalized Polygons ◽  
Completely Reducible ◽  
Standard Module ◽  
Rank 2

Suppose that (P,B,F) is a triple consisting of the points, blocks and flags of a generalized m-gon, and H(F) the associated rank-2 Iwahori–Hecke algebra. H(F) acts naturally on the integral standard module ZF based on F. This work gives arithmetic conditions on a subring R, where R contains the integers and is contained in the rationals, that insure the associated R-ary Iwahori–Hecke algebra to be completely reducible on RF. The constituent multiplicities are related to the R-normal form of the incidence matrix of (P,B,F).

scholarly journals HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS

2015 ◽  
Vol 16 (2) ◽  
pp. 351-419 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Paul Baum ◽  
Roger Plymen ◽  
Maarten Solleveld
Keyword(s):  
Finite Group ◽  
Local Field ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Linear Groups ◽  
Special Linear Groups ◽  
Affine Hecke Algebra ◽  
Morita Equivalent ◽  
Affine Hecke Algebras ◽  
A Finite Group

Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

scholarly journals ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En

2009 ◽  
Vol 18 (02) ◽  
pp. 237-264 ◽  
Author(s):  
R. M. GREEN
Keyword(s):  
Hecke Algebras ◽  
Markov Trace ◽  
Diagram Representation ◽  
Lusztig Polynomials

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.

scholarly journals MARKOV TRACE ON THE YOKONUMA–HECKE ALGEBRA

2004 ◽  
Vol 13 (01) ◽  
pp. 25-39 ◽  
Author(s):  
J. JUYUMAYA
Keyword(s):  
Hecke Algebra ◽  
Markov Trace

It is proved that the Yokonuma-Hecke algebra supports a Markov trace.

scholarly journals R-POLYNOMIALS OF FINITE MONOIDS OF LIE TYPE

2010 ◽  
Vol 20 (06) ◽  
pp. 793-805 ◽  
Author(s):  
KÜRŞAT AKER ◽  
MAHIR BILEN CAN ◽  
MÜGE TAŞKIN
Keyword(s):  
Weyl Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Möbius Function ◽  
Necessary Condition ◽  
Finite Monoid ◽  
Finite Monoids ◽  
Mobius Function ◽  
Renner Monoid

This paper studies the combinatorics of the orbit Hecke algebras associated with W × W orbits in the Renner monoid of a finite monoid of Lie type, M, where W is the Weyl group associated with M. It is shown by Putcha in [12] that the Kazhdan–Lusztig involution [6] can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the R-polynomials, we calculate the Möbius function of the Bruhat–Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group.

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