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

Abstract

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...