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R. M. Green

Publications and source records attributed to R. M. Green.

50 records · Page 3Linked to original sources
arXiv2002

On planar algebras arising from hypergroups

Let $A$ be an associative algebra with identity and with trace. We study the family of planar algebras on 1-boxes that arise from $A$ in the work of Jones, but with the added assumption that the labels on the 1-boxes come from a discrete hypergroup in the sense of Sunder. This construction equips the algebra $P_n^A$ with a canonical basis, $\BB_n^A$, which turns out to be a ``tabular'' basis. We examine special cases of this construction to exhibit a close connection between such bases and Kazhdan--Lusztig bases of Hecke algebras of types $A$, $B$, $H$ or $I$.

math.QA↗
arXiv2002

Categories arising from tabular algebras

We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a knowledge of the structure constants. This problem is equivalent to understanding a certain category (the category of table data associated to a tabular algebra) which we introduce. The main result is that this category is equivalent to another category (the category of based posets associated to a tabular algebra) whose structure we describe explicitly.

math.QA↗
arXiv2001

On 321-avoiding permutations in affine Weyl groups

We introduce the notion of 321-avoiding permutations in the affine Weyl group $W$ of type $A_{n-1}$ by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in $W$ coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of $W$ (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations. Using Shi's characterization of the Kazhdan--Lusztig cells in the group $W$, we use our main result to show that the fully commutative elements of $W$ form a union of Kazhdan--Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan--Lusztig basis of the associated Hecke algebra to be computed combinatorially. We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to $GL_n({\Bbb C})$.

math.CO↗
arXiv2001

Decorated tangles and canonical bases

We study the combinatorics of fully commutative elements in Coxeter groups of type $H_n$ for any $n > 2$. Using the results, we construct certain canonical bases (i.e., IC bases) for non-simply-laced generalized Temperley--Lieb algebras and show how to relate them to morphisms in the category of decorated tangles.

math.QA↗
arXiv2001

Tabular algebras and their asymptotic versions

We introduce tabular algebras, which are simultaneous generalizations of cellular algebras (in the sense of Graham-Lehrer) and table algebras (in the sense of Arad-Blau). We show that if a tabular algebra is equipped with a certain kind of trace map then the algebra has a corresponding asymptotic version whose structure can be explicitly determined. We also study various natural examples of tabular algebras.

math.QA↗
arXiv2001

Fully commutative Kazhdan-Lusztig cells

We investigate the compatibility of the set of fully commutative elements of a Coxeter group with the various types of Kazhdan-Lusztig cells using a canonical basis for a generalized version of the Temperley-Lieb algebra.

math.QA↗
arXiv1999

Canonical Bases for Hecke Algebra Quotients

We establish the existence of an IC basis for the generalized Temperley--Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.

math.QA↗
arXiv1999

Completions of Cellular Algebras

We introduce procellular algebras, so called because they are inverse limits of finite dimensional cellular algebras as defined by Graham and Lehrer. A procellular algebra is defined as a certain completion of an infinite dimensional cellular algebra whose cell datum is of ``profinite type''. We show how these notions overcome some known obstructions to the theory of cellular algebras in infinite dimensions.

math.QA↗
arXiv1998

On representations of affine Temperley-Lieb algebras, II

We study some non-semisimple representations of affine Temperley--Lieb algebras and related cellular algebras. In particular, we classify extensions between simple standard modules. Moreover, we construct a completion which is an infinite dimensional cellular algebra.

math.RT↗
arXiv1998

The Affine q-Schur algebra

We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of type $\hat A_{r-1}$, where $n \geq r$. This generalizes the original $q$-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinary $q$-Schur algebra and the affine Hecke algebra as subalgebras. Using this approach we can prove a double centralizer property. The second construction realizes the affine $q$-Schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module, analogous to the construction of the ordinary $q$-Schur algebra as a quotient of $U(\frak g \frak l_n)$.

q-alg↗
arXiv1997

Generalized Temperley-Lieb algebras and decorated tangles

We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley-Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and $D$. This generalizes Kauffman's diagram calculus for the Temperley-Lieb algebra.

q-alg↗
arXiv1997

Cellular algebras arising from Hecke algebras of type H_n

We study a finite-dimensional quotient of the Hecke algebra of type $H_n$ for general $n$, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring.

q-alg↗
arXiv1997

On the affine Temperley-Lieb algebras

We describe the cell structure of the affine Temperley-Lieb algebra with respect to a monomial basis. We construct a diagram calculus for this algebra.

q-alg↗