Quantum Super Littlewood Correspondences

Abstract: In this paper, we study the Littlewood theory associated with the quantum super immanants and supersymmetric polynomials, including both the super case and the quantum generalization. In the setting of quantum super Schur-Weyl duality between the quantum superalgebra $U_q(\mathfrak{gl}<em>{m|n})$ and the Iwahori-Hecke algebra $\mathcal{H}_r$ of type A, we explicitly construct basis vectors of the $(U_q(\mathfrak{gl}</em>{m|n}), \mathcal{H}<em>r)$-bimodule on the tensor product space $(\mathbb{C}^{{m|n})^{\otimes} r}$. Using this construction, we interpret the quantum super immanants via weight spaces of covariant tensor representations of $U_q(\mathfrak{gl}</em>{m|n})$.

• The paper establishes all three Littlewood correspondences for quantum supermatrices by replacing Schur polynomials with Schur supersymmetric functions and ordinary immanants with Hecke-character quantum super immanants.
• Quantum super immanants admit a weight-space interpretation as normalized supertraces under quantum super Schur–Weyl duality, and vanish for partitions outside the hook condition \(\lambda_{m+1}\le n\).
• The results yield quantum super analogs of the MacMahon Master Theorem, Newton, Goulden–Jackson, and Littlewood–Merris–Watkins identities, while the general quantum super Cayley–Hamilton theorem remains open beyond the \(m=n=1\) case.

Overview

This paper by Jing, Liu, and Zhang extends Littlewood's classical correspondences between Schur polynomials and immanants to the setting of quantum supermatrices. The work unifies two prior threads of the authors' program: the quantum (non-super) Littlewood correspondences for the quantum coordinate algebra, and the purely super case at $q \to 1$. The main objects are the quantum coordinate superalgebra $A_q(Mat_{m|n})$ generated by the RTT relations with the super $R$-matrix of type $A(m-1,n-1)$, and quantum super immanants defined via characters of the Iwahori-Hecke algebra $\mathcal{H}_r$ acting on tensor space $(\mathbb{C}^{m|n})^{\otimes r}$. The paper establishes all three Littlewood correspondences in this generality, a weight-space interpretation of quantum super immanants generalizing Kostant's trace formula, and derives as corollaries quantum super analogs of the MacMahon Master Theorem, Newton identities, Goulden-Jackson identities, and the Littlewood-Merris-Watkins identities.

Quantum super immanants

The authors define the quantum super immanant attached to an $\mathcal{H}_r$-representation $V$ with character $\chi^V$ by

$$Imm_{\chi^V}(X^I_J) = (-1)^{\sum_k \bar{i}_k \bar{j}_k}\langle i_1,\dots,i_r \mid \chi^V X_1\cdots X_r \mid j_1,\dots,j_r\rangle,$$

where $A_q(Mat_{m|n})$0 is expressed through primitive idempotents of $A_q(Mat_{m|n})$1, computed either from Young's orthogonal form or via the fusion procedure. A key structural result (Proposition 4.1) shows that for ordered multisets $A_q(Mat_{m|n})$2, the normalized immanant satisfies an averaging identity over the symmetric group, and — importantly — that $A_q(Mat_{m|n})$3 whenever $A_q(Mat_{m|n})$4, i.e., whenever $A_q(Mat_{m|n})$5. This vanishing is the super-analog of the classical constraint on partitions indexing nonzero covariant tensor representations, and it is proved using the fusion procedure for idempotents together with the RTT relations. The proof relies on the generic-$A_q(Mat_{m|n})$6 semisimplicity of $A_q(Mat_{m|n})$7; behavior at roots of unity is not addressed.

Weight space interpretation via Schur-Weyl duality

Using the quantum super Schur-Weyl duality of Moon and Mitsuhashi, the tensor space decomposes multiplicity-freely as $A_q(Mat_{m|n})$8. The paper combines Gelfand-Tsetlin bases of the covariant modules $A_q(Mat_{m|n})$9 (with explicit generator actions recalled from Palev-Stoilova-Van der Jeugt, Molev, Lu, and others) with Young's orthonormal basis of $R$0 to construct explicit bimodule basis vectors. Lemma 5.2 gives the precise dictionary: up to normalization by the Schur element $R$1 and the factor $R$2, the vector $R$3 is a multiple of $R$4 when the tableau substitution $R$5 is semistandard, and zero otherwise. The constants are normalized so that squared norms sum to one when several standard tableaux map to the same supertableau.

The central result of this section (Theorem 4.3) is the quantum super analog of Kostant's interpretation of immanants:

$R$6

i.e., the normalized quantum super immanant equals the supertrace of the coaction composed with projection onto the weight space indexed by the composition $R$7. This replaces Kostant's 0-weight-space trace formula for $R$8 with a full weight-space statement for $R$9, and it is the technical engine behind the correspondences that follow. The quantum Berezinian enters through its central character on Gelfand-Tsetlin bases, providing the scalar eigenvalues used implicitly in the weight decomposition.

Correspondences I and II

Theorems 5.1 and 5.2 establish the first two Littlewood correspondences: any polynomial relation among Schur supersymmetric polynomials translates into the corresponding relation among normalized quantum super immanants of principal minors (or complementary principal minors, summed over disjoint subsets). The proof proceeds by expanding induced representations $A(m-1,n-1)$0, whose characters decompose with Littlewood-Richardson coefficients $A(m-1,n-1)$1, and showing that the immanant of the induced character factors as a product of immanants over disjoint multisets divided by the factors $A(m-1,n-1)$2 — exactly mirroring the product rule for Schur supersymmetric polynomials. As a special case the authors obtain the quantum super Littlewood-Merris-Watkins identities relating immanants of induced sign and trivial characters to products of elementary and complete symmetric immanants.

Correspondence III and the commutative subalgebra

The third correspondence requires substitutes for characteristic roots, which do not exist for quantum supermatrices. The authors instead work with the commutative subalgebra $A(m-1,n-1)$3 generated by the elements $A(m-1,n-1)$4 (immanants of the trivial Hecke character), whose pairwise commutativity follows from the construction. The MacMahon Master Theorem $A(m-1,n-1)$5 yields Newton identities for the generating series, and the quantum super Goulden-Jackson identities (Theorem 5.6) state that both Jacobi-Trudi determinants built from $A(m-1,n-1)$6 and $A(m-1,n-1)$7 equal $A(m-1,n-1)$8, i.e., the summed normalized immanants.

The characteristic function $A(m-1,n-1)$9 then plays the role of the Berezinian-generated characteristic polynomial. Lemma 5.7 proves that the equation $\mathcal{H}_r$0 has a unique solution $\mathcal{H}_r$1 over an algebraic closure of the fraction field of the domain generated by the $\mathcal{H}_r$2; nonvanishing of the relevant Hankel-type determinant follows precisely from the Goulden-Jackson identities applied at $\mathcal{H}_r$3, while vanishing of the augmented determinants needed for consistency at lower degrees follows from Proposition 4.1's vanishing statement. With these "quantum super eigenvalues" in hand, Theorem 5.9 establishes correspondence III:

$\mathcal{H}_r$4

expressing each Schur supersymmetric polynomial evaluated at these abstract spectral elements as a sum of normalized quantum super immanants. Specializations give $\mathcal{H}_r$5, $\mathcal{H}_r$6, and the power-sum-type element $\mathcal{H}_r$7 as supersymmetric polynomials in the same arguments. Theorem 5.10 further identifies the restriction of the diagonal evaluation homomorphism $\mathcal{H}_r$8 as an isomorphism $\mathcal{H}_r$9, so that the normalized immanants indexed by $(\mathbb{C}^{m|n})^{\otimes r}$0 form a basis of the commutative subalgebra. Finally, Theorem 5.11 relates these sums to classical immanants of a lower Hessenberg matrix built from the $(\mathbb{C}^{m|n})^{\otimes r}$1, via Cramer's rule and Newton identities.

Limitations and open questions

Several qualifications are stated explicitly in the paper. First, the analysis assumes generic $(\mathbb{C}^{m|n})^{\otimes r}$2, under which $(\mathbb{C}^{m|n})^{\otimes r}$3 is semisimple; root-of-unity phenomena are not treated. Second, the elements $(\mathbb{C}^{m|n})^{\otimes r}$4 live only in an algebraic closure of a fraction field, not in $(\mathbb{C}^{m|n})^{\otimes r}$5 itself, so correspondence III is a statement over an extended coefficient ring. Third, the quantum super Cayley-Hamilton theorem is conjectural: the authors verify it only for $(\mathbb{C}^{m|n})^{\otimes r}$6, where they compute explicitly $(\mathbb{C}^{m|n})^{\otimes r}$7 and $(\mathbb{C}^{m|n})^{\otimes r}$8, and confirm $(\mathbb{C}^{m|n})^{\otimes r}$9; the general identity is known at $\mathcal{H}_r$0 (Urrutia-Morales) and at $\mathcal{H}_r$1 (Zhang), but remains open in the quantum super case. Whether the general Cayley-Hamilton identity holds — and whether the spectral elements admit a more intrinsic description inside the coordinate superalgebra — are the natural open questions left by this work.

Conclusion

The paper completes the Littlewood correspondence program for quantum supermatrices: all three of Littlewood's correspondences hold verbatim once Schur polynomials are replaced by Schur supersymmetric polynomials, ordinary immanants by Hecke-character immanants of the generator matrix of $\mathcal{H}_r$2, and characteristic roots by the unique solutions of the Berezinian-generated characteristic equation. The weight-space formula generalizing Kostant's trace identity provides a uniform representation-theoretic foundation, and the classical, super, and quantum cases are all recovered as specializations ($\mathcal{H}_r$3, $\mathcal{H}_r$4, respectively). The main outstanding problem is the general quantum super Cayley-Hamilton theorem, currently supported only by the $\mathcal{H}_r$5 computation.