Symmetric functions in noncommuting variables in superspace

Symmetric functions are fundamental objects with deep
applications in several areas of mathematics and physics \cite{Mac}. A
symmetric function is a formal power series $f(x_1,x_2,\ldots)$ that
remains invariant under any permutation of its variables.

In recent years, a new class of symmetric functions has been
introduced, involving the usual commuting variables
$x=(x_1,x_2,\ldots)$ together with anticommuting variables
$\theta=(\theta_1,\theta_2,\ldots)$. These are known as symmetric
functions in superspace. The theory of symmetric functions in
superspace has led to significant advances, extending many classical
results from the theory of symmetric functions
\cite{DLM1,DLM2,DLM3,JL}.

In this talk, we introduce a new family of symmetric functions
involving noncommuting variables together with anticommuting
variables, equipped with a diagonal action on both sets of variables.
Our results extend several aspects of the theory developed in
\cite{RoSa}, and the resulting functions are closely related to
symmetric functions in superspace. The theory is governed by the
combinatorics of set superpartitions, which index the natural bases.
This provides a concrete combinatorial framework underlying the
noncommutative superspace setting.

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