The Grothendieck ring of the periplectic Lie supergroup and supersymmetric functions

TL;DR

This paper establishes an isomorphism between the Grothendieck ring of finite-dimensional representations of the periplectic Lie supergroup and a specific ring of symmetric polynomials, revealing structural insights into supersymmetric functions.

Contribution

It provides a novel algebraic characterization of the Grothendieck ring for the periplectic Lie supergroup in terms of symmetric polynomials with a particular evaluation property.

Findings

Grothendieck ring is isomorphic to a ring of symmetric Laurent polynomials

Evaluation at specific points is independent of the parameter t

Connects representation theory of supergroups with symmetric functions

Abstract

We show that the Grothendieck ring of finite-dimensional representations of the periplectic Lie supergroup $P(n)$ is isomorphic to the ring of symmetric polynomials in $x_1^{\pm 1}, \ldots, x_n^{\pm 1}$ whose evaluation $x_1=x_2^{-1}=t$ is independent of $t$.