# Grothendieck rings of periplectic Lie superalgebras

## TL;DR

This paper explicitly describes the Grothendieck rings of finite-dimensional representations of periplectic Lie superalgebras, revealing their structure as symmetric polynomial rings with specific evaluation properties.

## Contribution

It provides an explicit description of the Grothendieck rings for periplectic Lie superalgebras, including the isomorphism to a symmetric polynomial ring with particular evaluation invariance.

## Key findings

- Grothendieck ring of $P(n)$ is isomorphic to symmetric polynomials in $x_i^{\u2212 1}$
- Evaluation at $x_1=x_2^{-1}=t$ is independent of $t$
- Explicit algebraic structure of representation rings for periplectic Lie superalgebras

## Abstract

We describe explicitly the Grothendieck rings of finite-dimensional representations of the periplectic Lie superalgebras. In particular, the Grothendieck ring of the 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 $t$.