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Path tableaux and combinatorial interpretations of immanants for class functions on $S_n$

Abstract : Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$.
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Sam Clearman, Brittany Shelton, Mark Skandera. Path tableaux and combinatorial interpretations of immanants for class functions on $S_n$. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.233-244. ⟨hal-01215097⟩

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