Factorial characters of classical Lie groups and their combinatorial realisations

TL;DR

This paper introduces factorial irreducible characters of classical Lie groups, providing determinant-based definitions, Jacobi-Trudi identities, and combinatorial tableau models for $gl(n)$, $so(2n+1)$, $sp(2n)$, and $o(2n)$, including new extensions for $so(2n)$.

Contribution

It extends factorial character theory to classical Lie groups with explicit determinant formulas, identities, and combinatorial models, including new cases for $so(2n)$.

Findings

Derived flagged Jacobi-Trudi identities for factorial characters.

Established combinatorial tableau models via non-intersecting lattice paths.

Extended factorial difference characters to $so(2n)$.

Abstract

Just as the definition of factorial Schur functions as a ratio of determinants allows one to show that they satisfy a Jacobi-Trudi-type identity and have an explicit combinatorial realisation in terms of semistandard tableaux, so we offer here definitions of factorial irreducible characters of the classical Lie groups as ratios of determinants that share these two features. These factorial characters are each specified by a partition, $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$, and in each case a flagged Jacobi-Trudi identity is derived that expresses the factorial character as a determinant of corresponding factorial characters specified by one-part partitions, $(m)$, for which we supply generating functions. These identities are established by manipulating determinants through the use of certain recurrence relations derived from these generating functions. The transitions to combinatorial realisations of the factorial characters in terms of tableaux are then established by means of non-intersecting lattice path models. The results apply to $gl(n)$, $so(2n+1)$, $sp(2n)$ and $o(2n)$, and are extended to the case of $so(2n)$ by making use of newly defined factorial difference characters.