We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$ representations are parameterized by pairs of Young diagrams $(λ,\barλ')$, where $\barλ'$ is the complement conjugate diagram to $λ$ inside the $n\times k$ rectangle. We set the probability of a diagram as a normalized specialization of the character for the corresponding irreducible component. For the principal specialization we get the probability that is equal to the ratio of the $q$-dimension for the irreducible component over the $q$-dimension of the exterior algebra. We demonstrate that this probability distribution can be described by the q-Krawtchouk polynomial ensemble. We derive the limit shape and prove the central limit theorem for the fluctuations in the limit when $n,k$ tend to infinity and $q$ tends to one at comparable rates.
We study large random partitions boxed into a rectangle and coming from skew Howe duality, or alternatively from dual Schur measures. As the sides of the rectangle go to infinity, we obtain: 1) limit shape results for the profiles generalizing the Vershik--Kerov--Logan--Shepp curve; and 2) universal edge asymptotic results for the first parts in the form of the Tracy--Widom distribution, as well as less-universal critical regime results introduced by Gravner, Tracy and Widom. We do this for a large class of Schur parameters going beyond the Plancherel or principal specializations previously st…
We extend a quantized skew Howe duality result for Type $\mathbf{A}$ algebras to orthogonal types via a seesaw. We develop an operator commutant version of the First Fundamental Theorem of invariant theory for $U_q(\mathfrak{so}_n)$ using a double centralizer property inside a quantized Clifford algebra. We obtain a multiplicity-free decomposition of tensor powers of the $U_q(\mathfrak{so}_{2n})$ spin representation by explicitly computing joint highest weights with respect to an action of $U_q(\mathfrak{so}_{2n}) \otimes U_q'(\mathfrak{so}_m)$. Clifford algebras are an essential feature of ou…
We describe how certain cyclotomic Nazarov-Wenzl algebras occur as endomorphism rings of projective modules in a parabolic version of BGG category O of type $D$. Furthermore we study a family of subalgebras of these endomorphism rings which exhibit similar behaviour to the family of Brauer algebras even when they are not semisimple. The translation functors on this parabolic category O are studied and proven to yield a categorification of a coideal subalgebra of the general linear Lie algebra. Finally this is put into the context of categorifying skew Howe duality for these subalgebras.
We consider the decomposition into irreducible components of the external power $Λ^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(λ,μ)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$μ=λ'$. We show that the Young diagram $λ$ which corresponds to a randomly selected irreducible component $(λ,λ')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular …
Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality
Université d’Angers（昂热大学）；St. Petersburg State University（圣彼得堡国立大学）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京应用数学与交叉科学研究院）；Hokkaido University（北海道大学）
The crystals for a finite-dimensional complex reductive Lie algebra $\mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{\mathfrak{g}}$, constructed using the Dynkin diagram of $\mathfrak{g}$, and its combinatorial action on any $\mathfrak{g}$-crystal via Schützenberger involutions. We compare this action with that of the Berenstein-Kirillov group on Gelfand-Tsetlin patterns. Henriques and Kamnitzer define an action of $C_n=C_{\mathfrak{gl}_n}$ on $n$-tensor products of $\mathfrak{g}$-crystals, for …
In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying the characters. The purpose of the current paper is to provide bijective proofs of such factorizations. The quantities involved have known combinatorial interpretations in terms of Gelfand-Tsetlin…
