A Plactic Algebra Action on Bosonic Particle Configurations
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A Plactic Algebra Action on Bosonic Particle Configurations Joanna Meinel1 Received: 6 July 2020 / Accepted: 15 October 2020 / © The Author(s) 2020
Abstract We study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra slN . It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra. Keywords Plactic algebra · Bosonic particle configurations · Kashiwara crystals · Center · Normal form Mathematics Subject Classification (2010) 05E10 (81R10, 17B37, 20G42)
Introduction The plactic monoid defined by Knuth relations in [13] appears in many different guises in combinatorics and representation theory. In particular it arises from Kashiwara crystals [10] of the quantum analogue of the special linear Lie algebra slN (C) and their realisation in terms of semistandard Young tableaux, see [7, Section 2.1] and [9] for background, details and references: The plactic monoid is isomorphic to the monoid of semistandard Young tableaux with entries 1, . . . , N − 1 and multiplication defined by row bumping. Relations among Kashiwara crystal operators are usually different and more difficult to describe. For abstract crystals of simply laced finite and affine type these relations were studied e.g. by Stembridge in [18] where a list of relations is given which are necessary and sufficient for the abstract crystal graph to be a crystal graph of an integrable highest weight representation. We are particularly interested in the crystal B (kω1 ) of the quantum analogue for the symmetric representation Symk (CN ) of slN (C). Here it is known that the Kashiwara crystal Presented by: Peter Littelmann Joanna Meinel
[email protected] 1
Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Bonn, Germany
J. Meinel
operators fi satisfy the Knuth relations together with the additional commutativity relation fi fj = fj fi for |i − j | > 1, see [5, Example 2.6]. Fomin and Greene develop a theory of Schur functions in noncommutative variables that applies in particular to the local plactic algebra (the semigroup algebra for the plactic monoid with the additional commutativity relation), including a generalized LittlewoodRichardson rule for Schur functions defined over the local plactic algebra. The local plactic algebra PN can be defined directly by generators and relations to be the unital associative k-algebra generated by a1 , . . . , aN−1 for N ≥ 3 over some ground field k subject to the plactic relations ai ai−1 ai = ai ai ai−1 for 2 ≤ i ≤ N − 1, ai ai+1 ai = ai+1 ai ai together with the commutativity relation a i aj = aj ai
for 1 ≤ i ≤ N − 2, for |i − j | > 1.
Accord
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