Folding KLR algebras

McNamara, Peter J.

doi:10.1112/jlms.12218

Cited by 3 publications

(1 citation statement)

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“…[, §§ 2.5.1]). The property (v) holds because the involution

*

is induced from the twist of

R_{d}

‐modules by the involutive automorphism

σ

R_{d}

in [, §§ 2.1] (see also [, Section 12]).…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 99%

A comparison of Newton–Okounkov polytopes of Schubert varieties

Fujita

Oya

2017

Journal of London Math Soc

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A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (resp., the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (resp., the Nakashima-Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott-Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (resp., by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation.Comment: 21 pages, to appear in J. London Math. Soc. (2

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“…[, §§ 2.5.1]). The property (v) holds because the involution

*

is induced from the twist of

R_{d}

‐modules by the involutive automorphism

σ

R_{d}

in [, §§ 2.1] (see also [, Section 12]).…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 99%

A comparison of Newton–Okounkov polytopes of Schubert varieties

Fujita

Oya

2017

Journal of London Math Soc

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Foldings of KLR algebras

Ma,

Shoji,

Zhou

2024

Journal of Algebra

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Folding with Soergel bimodules

Elias¹

2017

Categorification and Higher Representation Theory

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We give a gentle introduction to the concept of folding. That is, we provide an elementary discussion of equivariant categories, their weighted Grothendieck groups, and the technical aspects of computing with them. We then perform the computations required to confirm that quasi-split Hecke algebras with unequal parameters are categorified by equivariant Soergel bimodules, in almost every case.Remark 1.4. There is an action of /2 on the Weyl group of type F 4 , with invariant subgroup I 2 (8). We have not even attempted to perform the computations here. This missing case should follow by analogous computations. Regardless, the result has already been proven by Lusztig.Again, we do rely heavily on the truth of the Soergel conjecture for our categorification result. However, we do not use any of the Hodge-theoretic machinery required to prove the Soergel conjecture in [6]. Moreover, our computations do not rely on the Soergel conjecture, and apply in more generality.

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Folding KLR algebras

Abstract: 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.

Cited by 3 publications

References 28 publications

A comparison of Newton–Okounkov polytopes of Schubert varieties

A comparison of Newton–Okounkov polytopes of Schubert varieties

Foldings of KLR algebras

Folding with Soergel bimodules

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