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McNamara B. R. | McNamara Brian R. | Mcnamara Sean | McNamara Peter R. W. | McNamara Brian | McNamara Peter J. | McNamara B. | McNamara S. | McNamara Daniel J. | McNamara P. W.

Peter Patrick | Peter Annika H. G. | Peter Hardi | Peter H. | Peter P. | Péter M. | Péter L. | Peter Andreas | Peter David | Peter A. H. G.

J | J G. | J S. | J Girish Raguvir | J Ganesh | J Jeya Pradha | J Metilda Sagaya Mary N | J Paul A. Schweitzer S. | J Prabuchandran K. | J Abhijit A

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Singularities of Schubert varieties within a right cell

Lanini Martina, McNamara Peter J.
19 Mar 2020 math.RT math.AG math.CO arxiv.org/abs/2003.08616

We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in finding examples of reducible associated varieties of $\mathfrak{sl}_n$-highest weight modules, as well as in the study of $W$-graphs for symmetric groups, and in comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. For example, we are able to systematically produce many negative answers to a question from the 1980s of Borho-Brylinski and Joseph, which had been settled by Williamson via computer calculations only in 2014.

Non-perverse parity sheaves on the flag variety

McNamara Peter J.
01 Dec 2018 math.RT math.AG arxiv.org/abs/1812.00178

We give examples of non-perverse parity sheaves on Schubert varieties for all primes.

Monoidality of Kato's Reflection Functors

McNamara Peter J.
30 Nov 2017 math.RT math.QA arxiv.org/abs/1712.00173

Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal.

Representation theory of geometric extension algebras

McNamara Peter J.
27 Jan 2017 math.RT arxiv.org/abs/1701.07949

We study the question of when geometric extension algebras are polynomial quasihereditary. Our main theorem is that under certain assumptions, a geometric extension algebra is polynomial quasihereditary if and only if it arises from an even resolution. We give an application to the construction of reflection functors for quiver Hecke algebras.

Folding KLR algebras

McNamara Peter J.
18 Mar 2016 math.QA math.RT arxiv.org/abs/1603.05768

This paper develops the theory of KLR algebras with a Dynkin diagram automorphism. This is foundational material intended to allow folding techniques in the theory of KLR algebras.

Face Functors for KLR Algebras

McNamara Peter J., Tingley Peter
14 Dec 2015 math.RT math.QA arxiv.org/abs/1512.04458

Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of cuspidal representations realize crystals for sub Kac-Moody algebras. Here we put that observation an a firmer categorical footing by exhibiting a functor between the category of representations of the KLR algebra for the sub Kac-Moody algebra and the category of cuspidal representations of the original KLR algebra.

Representations of Khovanov-Lauda-Rouquier algebras III: Symmetric Affine Type

McNamara Peter J.
27 Jul 2014 math.RT math.QA arxiv.org/abs/1407.7304

We develop the homological theory of KLR algebras of symmetric affine type. For each PBW basis, a family of standard modules is constructed which categorifies the PBW basis.

Homological properties of finite type Khovanov-Lauda-Rouquier algebras

Brundan Jonathan, Kleshchev Alexander, McNamara Peter J.
25 Oct 2012 math.RT math.QA arxiv.org/abs/1210.6900

We give an algebraic construction of standard modules (infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an `affine quasi-hereditary algebra.' In simply-laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.

Finite Dimensional Representations of Khovanov-Lauda-Rouquier algebras I: Finite Type

McNamara Peter J.
24 Jul 2012 math.RT math.QA arxiv.org/abs/1207.5860

We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we prove a conjecture describing the global dimension of these algebras.

Factorial Schur functions and the Yang-Baxter equation

Bump Daniel, McNamara Peter J., Nakasuji Maki
15 Aug 2011 math.CO math-ph math.MP arxiv.org/abs/1108.3087

Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mechanical system (six vertex model). The proof is based on the Yang-Baxter equation. There is a deformation parameter $t$ which may be specialized in different ways. If $t=-1$, then we recover the expression of the factorial Schur function as a ratio of alternating polynomials. If $t=0$, we recover the description as a sum over tableaux. If $t=\infty$ we recover a description of Lascoux that was previously considered by McNamara. We also are able to prove using the Yang-Baxter equation the asymptotic symmetry of the factorial Schur functions in the shift parameters. Finally, we give a proof using our methods of the dual Cauchy identity for factorial Schur functions. Thus using our methods we are able to give thematic proofs of many of the properties of factorial Schur functions.