A Vertex Model for Supersymmetric LLT Polynomials

Abstract

We describe a Yang-Baxter integrable vertex model, which can be realized as a degeneration of a vertex model introduced by Aggarwal, Borodin, and Wheeler. From this vertex model, we construct a certain class of partition functions that we show are essentially equal to the super ribbon functions of Lam. Using the vertex model formalism, we give proofs of many properties of these polynomials, namely a Cauchy identity and generalizations of known identities for supersymmetric Schur polynomials. LLT polynomials, super ribbon functions, and the Littlewood quotient mapThis section provides necessary background information and establishes some notation for the rest of this paper. First we venture into the world of tableaux on tuples of skew shapes, and we define coinversion LLT polynomials. Then we venture into the world of ribbon tableaux, and we define super ribbon function. Finally, we connect these two worlds via the Littlewood quotient map.

Colored vertex models and $k$-tilings of the Aztec diamond

“…Next we discuss several families of vertex models, which generalize the vertices introduced in Section 2. These vertex models were originally defined and studied in [6,12], although they can be realized as degenerations of vertex models introduced in [1] (see [12,Lemma 3.1]).…”

“…The main goal of this paper is to study superpositions of domino tilings of the Aztec diamond, by using recent ideas coming from the study of colored vertex models. It has recently been shown [1,6,7] that the LLT polynomials can be realized as a certain class of partition functions constructed from an integrable vertex model, and more recently the same was shown for the supersymmetric LLT polynomials (also called super ribbon functions) [8,12]. We will use the vertex models introduced in [12], which can also be realized as degenerations of a vertex model introduced in [1].…”

“…For k ≥ 1, we define a k-tiling of the Aztec diamond of rank m to be the superposition of k tilings, colored from 1 to k. These tilings are not independent; we define an interaction between the tilings of colors i < j to be a pair of dominos, one of color i and one of color j, of a certain form. By relating these tilings to the colored vertex models introduced in [12], we prove that (1 + t x i y j ).…”

“…• In Section 3.1, we discuss the k-color vertex models that were introduced in [6,12] to study (super) LLT polynomials. In Section 3.2, we relate the rows of vertices to (co-)interlacing k-tuples of partitions.…”

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We study k-tilings (k-tuples of domino tilings) of the Aztec diamond of rank m. We assign a weight to each k-tiling, depending on the number of dominos of certain types and the number of "interactions" between the tilings. Employing the colored vertex models introduced in [12] to study supersymmetric LLT polynomials, we compute the generating polynomials of the ktilings. We then prove some combinatorial results about k-tilings, including a bijection between k-tilings with no interactions and 1-tilings, and we compute the arctic curves of the tilings for t = 0 and t → ∞.

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