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Register a new userLakshmibai-Seshadri paths and non-symmetric Cauchy identity
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We give a simple crystal theoretic interpretation of the Lascouxs expansion of a non-symmetric Cauchy kernel $prod_{i+ jleq n+1}(1-x_iy_j)^{-1}$, which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.
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Let $lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $mathrm{QLS}(lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $lambda$, $mu$, we have an isomorphism $Theta : mathrm{QLS}(lambda + mu) rightarrow mathrm{QLS}(lambda) otimes mathrm{QLS}(mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the moments F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a product of two matrices, ultimately yielding a polynomial in q=p^d. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric functions. This problem is closely to the study of maximal caps.
Let $(V,omega)$ be an orthosympectic $mathbb Z_2$-graded vector space and let $mathfrak g:=mathfrak{gosp}(V,omega)$ denote the Lie superalgebra of similitudes of $(V,omega)$. When the space $mathscr P(V)$ of superpolynomials on $V$ is emph{not} a completely reducible $mathfrak g$-module, we construct a natural basis $D_lambda$ of Capelli operators for the algebra of $mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_lambda$ on maximal indecomposable components of $mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $mathscr P(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_lambda$ are emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators ${mathbf D_{t,lambda}}_{lambdainmathcal P}^{}$ that induce morphisms of indecomposable components of symmetric powers of $mathsf V_t$, where $mathsf V_t$ is the generating object of $mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the $left{mathbf D_{t,lambda}right}_{lambdainmathcal P}$ that are analogous to our results for the operators ${D_lambda}_{lambdainmathcal P}^{}$.
We determine the multiplicities of irreducible summands in the symmetric and the exterior squares of hook representations of symmetric groups over an algebraically closed field of characteristic zero.
We establish an isomorphism between the center of the Heisenberg category defined by Khovanov and the algebra $Lambda^*$ of shifted symmetric functions defined by Okounkov-Olshanski. We give a graphical description of the shifted power and Schur bases of $Lambda^*$ as elements of the center, and describe the curl generators of the center in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov and the noncommutative probability spaces of Biane.
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