Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras

Bowman, C.; Cox, Anton; Hazi, Amit

doi:10.48550/arxiv.2005.02825

Cited by 8 publications

(19 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2 Moreover, one might consider not just a unitary representation by itself, but the whole Serre subcategory or subquotient category determined by the saturated subset of the poset of lowest weights appearing in its character. In type A, this subcategory of O1 e (S n ) is equivalent to a truncation of an Elias-Williamson category of parabolic Soergel bimodules of type A e ′ ⊂ A e ′ for some e ′ ≤ e [3].…”

Section: Motivation and Summary Of Resultsmentioning

confidence: 99%

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

Norton¹

2020

Preprint

View full text Add to dashboard Cite

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations, showing that all possible supports allowed by the first and second results do in fact occur.

show abstract

Section: Motivation and Summary Of Resultsmentioning

confidence: 99%

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

Norton¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…. , j g ) (g = m − h), is given by (5.5.10) that is, t t t = t t t µ for µ = (1 (3) , 1 (3) , 1 (9) , 1 (3) ).…”

Section: An Optimized Set Of Generatorsmentioning

confidence: 99%

On Generalized blob algebras: Vertical idempotent truncations and Gelfand-Tsetlin subalgebras

Lobos¹

2022

Preprint

View full text Add to dashboard Cite

We study a family of idempotent truncations of the generalized blob algebra. Particularly we study the structure of the Gelfand-Tsetlin subalgebras of these algebras, obtaining optimal presentations in terms of generators and relations, monomial basis and dimension formula. 1 Introduction 1.1 Motivation Since their definition due to Martin and Saleur [18] and Martin and Woodcock [20] respectively, blob algebras and generalized blob algebras have received considerable amount of interest. It is known that their representation theory is strongly connected with Kazhdan-Lusztig polynomials in characteristic 0 and with p−Kazhdan-Lusztig polynomials in characteristic p (see [4], [20], [29], [15], [3]). Currently one of the central open problems in Representation Theory is the explicit calculus of p−Kazhdan-Lusztig polynomials since they are connected with an important spectrum of unsolved problems in different branches of pure mathematics, particularly they contain fundamental information of the modular representation theory of the symmetric group.Due to the work of Libedinsky and Plaza [15] and Bowman, Cox and Hazi, [3] we know that p−Kazhdan-Lusztig polynomials in type A can be obtained as graded decomposition numbers of generalized blob algebras. Therefore, in order to develop concrete strategies to obtain explicitly the p−Kazhdan-Lusztig polynomials coming from the world of blob algebras, it is necessary to understand more in detail the structure of generalized blob algebras. In this sense, this work aims to contribute to a better understanding of these algebras.Jucys-Murphy elements and the subalgebra that they generate have been considered fundamental objects in the study of representation theory of several important algebras as the group algebra of the symmetric group and associated Hecke algebras (see [26]), Ariki-Koike algebras (see [21]), Cyclotomic Hecke algebras (see [12]) among other. Moreover, due to the work of Mathas ([22]), we know that for a cellular algebra A, equipped with a family of Jucys-Murphy elements satisfying certain separation conditions, one can construct seminormal forms and explicitly compute Gram determinant of the irreducible modules or at least, under less exigent conditions, to obtain a decomposition of A, into a direct sum of smaller cellular subalgebras.Few years ago, Ryom-Hansen and the author ([17]) have explicitly built graded cellular basis for generalized blob algebras B m , provided with a set of Jucys-Murphy elements (in the sense of [22]). The main focus of this article is the study of the Gelfand-Tsetlin subalgebra of the generalized blob algebra, that is, the commutative subalgebra generated by the Jucys-Murphy elements obtained in [17]. We separate this study by working on the different idempotent truncations of B m . In this case we aboard a particular family of idempotent truncations that we call vertical and quasi-vertical. We obtain a complete description for these cases, providing optimal presentations in terms of generators and relations, monomial basis and a dimens...

show abstract

“…For λ ∈ P h (n), we identify Path h (λ) with the set of standard λ-tableaux in the obvious manner (see [BCH20] for more details). This identification preserves the grading.…”

Section: The Alcove Geometry Setmentioning

confidence: 99%

“…. , ε h ) and indeed this is the path used in [BCH20] (where it is implicitly assumed that ℓ is a step change). We further remark that one can always reorder the charge s ∈ Z ℓ to obtain some s ∈ Z ℓ for which ℓ is a step change (using the trivial algebra isomorphism H n (s) ∼ = H n ( s)).…”

Section: The Alcove Geometry Setmentioning

confidence: 99%

“…Sections 3 and 4 are devoted to the proof of Theorem B, which involves intricate combinatorial constructions. In Section 5 we recall previous work of the first author together with A. Cox and A. Hazi [BCH20] that will allow us to prove Theorem C. We do this in Section 6. In this section we also discuss the consequences of our work in the representation theory of rational Cherednik algebras.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions

Bowman¹,

Norton²,

Simental³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We relate the classes of unitary and calibrated representations of cyclotomic Hecke algebras and, in particular, we show that for the most important deformation parameters these two classes coincide. We classify these representations in terms of both multipartition combinatorics and as the points in the fundamental alcove under the action of an affine Weyl group. Finally, we cohomologically construct these modules via BGG resolutions.

show abstract

Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras

Cited by 8 publications

References 11 publications

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

On Generalized blob algebras: Vertical idempotent truncations and Gelfand-Tsetlin subalgebras

Unitary representations of cyclotomic Hecke algebras at roots of unity: combinatorial classification and BGG resolutions

Contact Info

Product

Resources

About