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109
Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers
 J. Funct. Anal
"... Abstract. We give asymptotic formulas for the multiplicities of weights and irreducible summands in hightensor powers V ⊗N λ of an irreducible representation Vλ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, rangin ..."
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Abstract. We give asymptotic formulas for the multiplicities of weights and irreducible summands in hightensor powers V ⊗N λ of an irreducible representation Vλ of a compact connected Lie group G. The weights are allowed to depend on N, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths with steps in a convex polytope. Contents
The FourierJacoby map and small representations, Representation Theory 7
 Department of Mathematics, University of Utah
, 2003
"... Abstract. We study the “FourierJacobi ” functor on smooth representations of split, simple, simplylaced padic groups. This functor has been extensively studied on the symplectic group, where it provides the representationtheoretic analogue of the FourierJacobi expansion of Siegel modular forms. ..."
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Cited by 11 (0 self)
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Abstract. We study the “FourierJacobi ” functor on smooth representations of split, simple, simplylaced padic groups. This functor has been extensively studied on the symplectic group, where it provides the representationtheoretic analogue of the FourierJacobi expansion of Siegel modular forms. Our applications are different from those studied classically with the symplectic group. In particular, we are able to describe the composition series of certain degenerate principal series. This includes the location of minimal and small (in the sense of the support of the local character expansion) representations as spherical subquotients.
Frontiers of reality in Schubert calculus
 Bulletin of the AMS
"... Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fu ..."
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Cited by 10 (5 self)
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Abstract. The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.
Moment zeta functions for toric CalabiYau hypersurfaces
 Communications in Number Theory and Physics, Vol
, 2007
"... Let n ≥ 2 be a positive integer. We consider the following family 1 Xλ: x1 + · · · + xn + = λ ..."
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Cited by 9 (6 self)
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Let n ≥ 2 be a positive integer. We consider the following family 1 Xλ: x1 + · · · + xn + = λ
COHOMOLOGICAL STRUCTURE OF THE MAPPING CLASS GROUP AND BEYOND
, 2005
"... In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer au ..."
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In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of free groups, the symplectomorphism group of surfaces. Based on these, we present several conjectures and problems concerning the cohomology of these groups. We are particularly interested in the possible interplays between these cohomology groups rather than merely the structures of individual groups. It turns out that, we have to include, in our considerations, two other groups which contain the mapping class group as their core subgroups and whose structures seem to be deeply related to that of the mapping class group. They are the arithmetic mapping class group and the group of homology cobordism classes of homology cylinders. 1
Geometric complexity theory and tensor rank, arXiv:1011.1350
, 2010
"... Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tens ..."
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Cited by 9 (5 self)
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Mulmuley and Sohoni [25, 26] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G = GL(W1) × GL(W2) × GL(W3) acting on the tensor product W = W1 ⊗ W2 ⊗ W3 of complex finite dimensional vector spaces. Let Gs = SL(W1) × SL(W2) × SL(W3). A key idea from [26] is that the irreducible Gsrepresentations occurring in the coordinate ring of the Gorbit closure of astabletensorw∈Ware exactly those having a nonzero invariant with respect to the stabilizer group of w. However, we prove that by considering Gsrepresentations, only trivial lower bounds on border rank can be shown. It is thus necessary to study Grepresentations, which leads to geometric extension problems that are beyond the scope of the subgroup restriction problems emphasized in [25, 26] and its follow up papers. We prove a very modest lower bound on the border rank of matrix multiplication tensors using Grepresentations. This shows at least that the barrier for Gsrepresentations can be overcome. To advance, we suggest the coarser approach to replace the semigroup of representations of a tensor by its moment polytope. We prove first results towards determining the moment polytopes of matrix multiplication and unit tensors. A full version of this paper is available at arxiv.org/abs/1011.1350
Connections Between The Representations Of The Symmetric Group And The Symplectic Group In Characteristic 2
 J. Algebra
"... this paper we show that each composition factor for the action of Sp 2l (K) on ..."
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this paper we show that each composition factor for the action of Sp 2l (K) on
New lower bounds for the border rank of matrix multiplication
"... Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Li ..."
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Abstract. The border rank of the matrix multiplication operator for n × n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n 2 − n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3 2 n2 + n 2 − 1 for all n ≥ 3. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.
The exterior algebra and “spin” of an orthogonal gmodule
 Transform. Groups
"... Let g be a reductive algebraic Lie algebra over an algebraically closed field k of characteristic zero and G is the corresponding connected and simply connected group. The symmetric algebra of a (finitedimensional) gmodule V is the algebra of polynomial ..."
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Cited by 7 (4 self)
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Let g be a reductive algebraic Lie algebra over an algebraically closed field k of characteristic zero and G is the corresponding connected and simply connected group. The symmetric algebra of a (finitedimensional) gmodule V is the algebra of polynomial
The Klein quartic in number theory
, 1999
"... Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � ..."
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Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat.