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

Abstract

International audience 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 $Imm_{{\phi} ^λ} (A)$. Soit $χ ^λ$ le caractère irréductible de $S_n$ qui correspond à la partition λ de l'entier n, ou de manière équivalente, la préimage de la fonction de Schur $s_λ$ par l'application caractéristique de Frobenius. Soit $\phi ^λ$ la fonction $S_n →ℂ$ qui est la préimage de la fonction symétrique monomiale m_λ . La valeur du caractère irréductible immanent $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ est non négative pour chaque matrice totalement non négative. Nous donnons une interprétation combinatoire de la valeur $Imm_λ (A)$ lorsque $λ$ est une partition en équerre. Stembridge a conjecturé que la valeur de l'immanent monomial $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ de $\phi ^λ$ est elle aussi non négative pour chaque matrice totalement non négative. Nous confirmons cette conjecture quand λ satisfait $λ _1 ≤2$, et nous donnons une interprétation combinatoire de $Imm_{{\phi} ^λ} (A)$ dans ce cas.

Evaluations of Hecke Algebra Traces at Kazhdan-Lusztig Basis Elements

“…We will let F w and G ′ w denote the zig-zag network and augmented star network corresponding to a fixed 3412-avoiding, 4231-avoiding permutation w, and we will let w(F ) denote the 3412avoiding, 4231-avoiding permutation corresponding to a fixed zig-zag network F . For example, suppose that F is the zig-zag network obtained from the concatenation G = G [3,7] • G [5,8] • G [8,9] • G [1,2] • G [2,4] of star networks of order 9. Drawing the poset on these intervals from left to right, we have .…”

“…Since the only covering pairs which intersect at more than an endpoint are [3,7] ≺• [5,8] and [3,7] ≺• [2,4], we construct G ′ by inserting G [3,7]∩ [5,8] = G [5,7] and G [3,7]∩ [2,4] = G [3,4] after G [3,7] . Thus we have G ′ = G [3,7] • G [5,7] • G [3,4] • G [5,8] • G [8,9] • G [1,2] • G [2,4] , and we obtain the 3412-avoiding, 4231-avoiding permutation w = w(F ) = 419763258: , w = 123456789 419763258 .…”

“…Note that we have G [3,4] • G [5,7] ∼ = G [5,7] • G [3,4] , since the intervals [3,4], [5,7] do not overlap. In general, we have the following.…”

“…We will always assume that path π i begins at source i. If for some w ∈ S n with one-line notation w 1 • • • w n , each component path π i terminates at sink w i , we will say that π has type w and we will write type(π) = w. If the union of the paths of π is equal to G, we will say that π covers G. For example, the planar network G [2,4] • G [1,3] can be covered by many different path families, including two of type e, and two of type 2341: (3.5) .…”

“…These will be discussed in Section 4. There is no conjectured combinatorial interpretation of φ λ (C ′ w (1)), even for w avoiding the patterns 3412 and 4231, but interpretations have been given for particular partitions λ by Stembridge [32], several of the authors [4], and Wolfgang [34]. The problem of interpreting θ(C ′ w (1)) when w does not avoid the patterns 3412 and 4231 is open.…”