Super-immanants and Littlewood correspondences

In this paper, we introduce the notion of super-immanants for supermatrices over a supercommutative algebra. Using the super Schur-Weyl duality we show that the super immanants play a significant role in covariant tensor representations of the general linear Lie superalgebra. Among various things, we obtain a supertrace formula for super-immanants, which generalizes Kostant’s trace formula to the super setting. Furthermore, we show that the Littlewood correspondences between super-immanants and supersymmetric polynomials establish an isomorphism between their corresponding algebras.

💡 Research Summary

The paper introduces a new class of matrix functions called “super‑immanants” for supermatrices whose entries lie in a super‑commutative algebra (for instance a Grassmann algebra). After setting up the basic notation—C^{m|n} as a Z₂‑graded vector space, the algebra of even endomorphisms Mat_{m|n}(R), and the coordinate super‑algebra A(Mat_{m|n}) generated by the entries x_{ij} with parity \bar i+\bar j—the authors define the super‑trace str and the multi‑trace str_a that will later appear in character formulas.

A representation of the symmetric group S_r on the r‑fold tensor power (C^{m|n})^{\otimes r} is constructed (formula (2.5)). The action is Z₂‑graded: each permutation acquires a sign depending on the parities of the permuted basis vectors. Using this representation they define the super‑immanant associated with any S_r‑module V (character χ_V) by the formula (2.7). This is a direct generalization of the classical immanant: it is a signed sum over permutations of products of matrix entries, weighted by the character of V, but now the signs also encode the super‑grading.

Proposition 2.1 is a cornerstone: for a multiset I of row (and column) indices and a partition λ of r that belongs to the “hook” set H(m,n;r) (i.e. λ_{m+1} ≤ n), the super‑immanant Imm_{χ_λ}(X_I) can be written as a matrix element of the primitive idempotent E_{λ,T} (associated with a standard Young tableau T) acting on the tensor basis. If λ lies outside the hook, the super‑immanant vanishes. The proof uses the Jucys‑Murphy elements and the fusion procedure for primitive idempotents, showing that any tensor containing more than m even or more than n odd basis vectors in a forbidden configuration is annihilated by E_{λ,T}. This reflects the intrinsic limitation of a super‑matrix: the even part has dimension m, the odd part n.

Section 2.2 reviews the super Schur‑Weyl duality (Berele‑Regev, Sergeev). The tensor space (C^{m|n})^{\otimes r} decomposes multiplicity‑free as a bimodule for U(gl_{m|n}) and S_r: \