The nil-blob algebra: An incarnation of type<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>Soergel calculus and of the truncated blob algebra

Lobos, Diego; Plaza, David; Ryom-Hansen, Steen

doi:10.1016/j.jalgebra.2020.11.021

Cited by 5 publications

(5 citation statements)

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“…This subalgebra plays an important role for what follows. To a certain extent, this runs parallel to several recent papers, for example [19] and [21], where similar idempotent truncated algebras have been studied. By general principles, eTL…”

Section: The Integral Klr-algebrasupporting

confidence: 67%

“…Statements similar to the one of the following Corollary, but for the original KLR-algebra R n defined over a field, are already present in literature, see for example [18], [21] and [19], although the proofs in these references are different from ours, since they rely on KLR-diagrammatics.…”

Section: The Integral Klr-algebramentioning

confidence: 78%

“…n e(i) that is obtained from S i by converting each s j to ψ j , and finally multiplying on the left and on the right by e. Similar elements have been considered before in [18], [19] and [21], but only for the original KLR-algebra R n defined over a field. In [19] and [21], the U i 's are called diamonds. For example, for n = 14 and p = 3 we have…”

Section: The Integral Klr-algebramentioning

confidence: 95%

“…(1.3) Similar elements have been already been considered before for example in [18], [19], [20] and [21], but only in the context of the original KLR-algebra R n defined over a field.…”

Section: Introductionmentioning

confidence: 92%

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Seminormal forms for the Temperley-Lieb algebra

Bastías¹,

Ryom-Hansen²

2023

Preprint

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Let TL Q n be the rational Temperley-Lieb algebra, with loop parameter 2. In the first part of the paper we study the seminormal idempotents Et for TL Q n for t running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of Et using Jones-Wenzl idempotents JW k for TL Q k where k ≤ n. In the second part of the paper we consider the Temperley-Lieb algebra TL Fp n over the finite field Fp, where p > 2. The KLR-approach to TL Fp n gives rise to an action of a symmetric group Sm on TL Fp n , for some m < n. We show that the Et's from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for Sm. This leads to a KLR-interpretation of the p-Jones-Wenzl idempotent p JWn for TL Fp n , that was introduced recently by Burull, Libedinsky and Sentinelli.

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Section: The Integral Klr-algebrasupporting

confidence: 67%

Section: The Integral Klr-algebramentioning

confidence: 78%

Section: The Integral Klr-algebramentioning

confidence: 95%

“…(1.3) Similar elements have been already been considered before for example in [18], [19], [20] and [21], but only in the context of the original KLR-algebra R n defined over a field.…”

Section: Introductionmentioning

confidence: 92%

See 2 more Smart Citations

Seminormal forms for the Temperley-Lieb algebra

Bastías¹,

Ryom-Hansen²

2023

Preprint

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“…Proof. In [16] there is a proof for certain particular cases in level l = 2. The general case is obtained analogously.…”

Section: Idempotent Truncations Of B Mmentioning

confidence: 99%

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

Lobos¹

2022

Preprint

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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...

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Seminormal forms for the Temperley-Lieb algebra

Ormeño Bastías,

Ryom-Hansen

2024

Journal of Algebra

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The nil-blob algebra: An incarnation of typeA˜1Soergel calculus and of the truncated blob algebra

Cited by 5 publications

References 20 publications

Seminormal forms for the Temperley-Lieb algebra

Seminormal forms for the Temperley-Lieb algebra

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

Seminormal forms for the Temperley-Lieb algebra

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