Supersymmetric Elements in Divided Powers Algebras
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Supersymmetric Elements in Divided Powers Algebras Frantiˇsek Marko1 Received: 22 December 2018 / Accepted: 15 October 2019 / © Springer Nature B.V. 2020
Abstract Description of adjoint invariants of general linear Lie superalgebras gl(m|n) by Kantor and Trishin was given in terms of supersymmetric polynomials. Later, generators of invariants of the adjoint action of the general linear supergroup GL(m|n) and generators of supersymmetric polynomials were determined over fields of positive characteristic. In this paper, we introduce the concept of supersymmetric elements in the divided powers algebra Div[x1 , . . . , xm , y1 , . . . , yn ], and give a characterization of supersymmetric elements via a system of linear equations. Then we determine generators of supersymmetric elements for divided powers algebras in the cases when n = 0, n = 1, and m ≤ 2, n = 2. Keywords Divided powers · Supersymmetric element · General linear supergroup · Adjoint invariants Mathematics Subject Classification (2010) Primary: 20G05; Secondary: 17A70 · 22E47
1 Introduction and Notation We start by recalling a description of the invariants of conjugacy classes of matrices, which is a classical problem in the invariant theory - see Chapter 1 of [4]. Let K be an infinite field of characteristic zero, V be a K-space of dimension m, E be the space of all K-linear maps of V (a choice of a basis of V identifies E with m × m-matrices). The general linear group GL(V ) acts on E via conjugation g.a = gag −1 for g ∈ GL(V ) and a ∈ E (corresponding to a change of basis of V ). Consider the space S(E) of all K-valued polynomial functions on E together with the action of GL(V ) given as g.f (a) = f (g −1 a) for g ∈ GL(V ), f ∈ S(E) and a ∈ E. By Chevalley’s restriction theorem (see Theorem 1.5.7 of [4]), the ring of invariants S(E)GL(V ) is isomorphic to the ring of symmetric polynomials K[x1 , . . . , xm ]. Here the variables x1 , . . . , xm are related to the coefficients σi (M) of the characteristic polynomial of m × m-matrices M. Presented by: Michel Brion. Frantiˇsek Marko
[email protected] 1
Pennsylvania State University, 76 University Drive, Hazleton, PA 18202, USA
F. Marko
The vertices of the Dynkin diagram = Am corresponding to GL(m) are given by simple positive roots that can be labeled by 1, . . . , m. The action of the Weyl group W permutes the elements of the set {1, . . . , m}. If we denote by P the ring of polynomials in variables x1 , . . . , xm , then the ring of symmetric polynomials P W consists of invariants of P under the action induced by the Weyl group W . The above classical correspondence was extended to the case of general linear supergroups GL(m|n) in characteristic zero by Kantor and Trishin in [2]. They have described the polynomial invariants of the adjoint action of the general linear Lie supergroup gl(m|n) and their connection to the algebra As of supersymmetric polynomials. The algebra As consists of polynomials f (x|y) = f (x1 , . . . , xm , y1 , . . . , yn ) that are symmetric in variables d f (x|y)|x1 =y1 =T =
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