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228
PBW filtration and bases for irreducible modules in type An, Transformation Groups: Volume 16, Issue 1
, 2011
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An invariant regarding Waring’s problem for cubic polynomials
"... to the memory of Michael Schneider, ten years after Abstract. We compute the equation of the 7secant variety to the Veronese variety (P 4,O(3)), its degree is 15. This is the last missing invariant in the AlexanderHirschowitz classification. It gives the condition to express a homogeneous cubic po ..."
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to the memory of Michael Schneider, ten years after Abstract. We compute the equation of the 7secant variety to the Veronese variety (P 4,O(3)), its degree is 15. This is the last missing invariant in the AlexanderHirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian. 1.
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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Cited by 16 (5 self)
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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
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 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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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.
Contact Projective Structures
 Indiana Univ. Math. J
"... Abstract. A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a onedimensional subspace of the contact distribution at that point there is a unique path of the family passing through the g ..."
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Abstract. A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a onedimensional subspace of the contact distribution at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with onedimensional fibers over the contact manifold and using this the local equivalence problem for contact projective structures is solved by the construction of a canonical regular Cartan connection. This Cartan connection is normal if and only if an invariant contact torsion vanishes. Every contact projective structure determines canonical paths transverse to the contact structure which fill out the contact projective structure to give a full projective structure, and the vanishing of the contact torsion implies the contact
COHOMOLOGY OF THE HILBERT SCHEME OF POINTS ON A SURFACE WITH VALUES IN REPRESENTATIONS OF TAUTOLOGICAL BUNDLES
, 2008
"... Let X a smooth quasiprojective algebraic surface, L a line bundle on X. Let X [n] the Hilbert scheme of n points on X and L [n] the tautological bundle on X [n] naturally associated to the line bundle L on X. We explicitely compute the image Φ(L [n]) of the tautological bundle L [n] for the Bridge ..."
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Let X a smooth quasiprojective algebraic surface, L a line bundle on X. Let X [n] the Hilbert scheme of n points on X and L [n] the tautological bundle on X [n] naturally associated to the line bundle L on X. We explicitely compute the image Φ(L [n]) of the tautological bundle L [n] for the BridgelandKingReid equivalence Φ: Db (X[n]) → Db Sn (Xn) in terms of a complex C • L of Snequivariant sheaves in Db Sn (Xn). We give, moreover, a characterization of the image Φ(L [n] ⊗ · · ·⊗L[n]) in terms of of the hyperderived spectral sequence E p,q 1 associated to the derived kfold tensor power of the complex C • L. The study of the Sninvariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general kfold exterior power of the tautological bundle for the HilbertChow morphism, providing DanilaBriontype formulas in these two cases. This yields easily the computation of the cohomology of X [n] with values in L [n] ⊗ L [n] and ΛkL [n].
PBWfiltration and bases for symplectic Lie algebras
 International Mathematics Research Notices 2011; doi: 10.1093/imrn/rnr014
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THE MODULI SPACE OF CUBIC THREEFOLDS VIA DEGENERATIONS OF THE INTERMEDIATE JACOBIAN
, 2007
"... A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds ..."
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A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of AllcockCarlsonToledo and LooijengaSwierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves.