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Register a new userLocalization and a generalization of MacDonalds inner product
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Erik Carlsson
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We find a limit formula for a generalization of MacDonalds inner product in finitely many variables, using equivariant localization on the Grassmannian variety, and the main lemma from cite{Car}, which bounds the torus characters of the higher c{C}ech cohomology groups. We show that the MacDonald inner product conjecture of type $A$ follows from a special case, and the Pieri rules section of MacDonalds book cite{Mac}, making this limit suitable replacement for the norm squared of one, the usual normalizing constant.
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We prove a Littlewood-Richardson type formula for $(s_{lambda/mu},s_{ u/kappa})_{t^k,t}$, the pairing of two skew Schur functions in the MacDonald inner product at $q = t^k$ for positive integers $k$. This pairing counts graded decomposition numbers in the representation theory of wreath products of the algebra $C[x]/x^k$ and symmetric groups.
This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal structure. This is work is motivated by defects in topological field theories.
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Edge estimation problem in unweighted graphs using local and sometimes global queries is a fundamental problem in sublinear algorithms. It has been observed by Goldreich and Ron (Random Structures & Algorithms, 2008), that weighted edge estimation for weighted graphs require $Omega(n)$ local queries, where $n$ denotes the number of vertices in the graph. To handle this problem, we introduce a new inner product query on matrices. Inner product query generalizes and unifies all previously used local queries on graphs used for estimating edges. With this new query, we show that weighted edge estimation in graphs with particular kind of weights can be solved using sublinear queries, in terms of the number of vertices. We also show that using this query we can solve the problem of the bilinear form estimation, and the problem of weighted sampling of entries of matrices induced by bilinear forms. This work is the first step towards weighted edge estimation mentioned in Goldreich and Ron (Random Structures & Algorithms, 2008).
Let $pi$ be a genuine cuspidal representation of the metaplectic group of rank $n$. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension $2n+1$. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands $L$-function of $pi$ twisted by a character. The bulk of this article focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature.
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