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64
J.Propp, The shape of a typical boxed plane partition
 J. of Math
, 1998
"... Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the ..."
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Cited by 51 (5 self)
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Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.
Weyl group multiple Dirichlet series II: The Stable Case
"... To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity tha ..."
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Cited by 24 (16 self)
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To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of nth order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.
On a GaussGivental Representation of Quantum Toda Chain Wave Function
"... We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of U(gl(N)) in terms of first order differential operators in Givental variables. The construction of this representation ..."
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Cited by 21 (10 self)
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We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of U(gl(N)) in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter Qoperator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.
Vertices of GelfandTsetlin polytopes
 Discrete Comput. Geom
"... This paper is dedicated to Louis Billera on the occasion of his sixtieth birthday. Abstract: This paper is a study of the polyhedral geometry of GelfandTsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertice ..."
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Cited by 12 (4 self)
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This paper is dedicated to Louis Billera on the occasion of his sixtieth birthday. Abstract: This paper is a study of the polyhedral geometry of GelfandTsetlin patterns arising in the representation theory gl nC and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowestdimensional face containing a given GelfandTsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov [1] about the integrality of all vertices of the GelfandTsetlin polytopes. We can construct for each n ≥ 5 a counterexample, with arbitrarily increasing denominators as n grows, of a nonintegral vertex. This is the first infinite family of nonintegral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the nonintegral vertices when n is fixed. 1
Representations of twisted Yangians associated with skew Young diagrams
 Selecta Math
"... To Professor I.M.Gelfand on his 90th birthday Abstract. Let GM be either the orthogonal group OM or the symplectic group SpM over the complex field; in the latter case the nonnegative integer M has to be even. Classically, the irreducible polynomial representations of the group GM are labeled by p ..."
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Cited by 11 (4 self)
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To Professor I.M.Gelfand on his 90th birthday Abstract. Let GM be either the orthogonal group OM or the symplectic group SpM over the complex field; in the latter case the nonnegative integer M has to be even. Classically, the irreducible polynomial representations of the group GM are labeled by partitions µ = (µ1, µ2,...) such that µ ′ 1 + µ ′ 2 � M in the case GM = OM, or 2µ ′ 1 � M in the case GM = SpM. Here µ ′ = (µ ′ 1, µ ′ 2,...) is the partition conjugate to µ. Let Wµ be the irreducible polynomial representation of the group GM corresponding to µ. Regard GN × GM as a subgroup of GN+M. Then take any irreducible polynomial representation Wλ of the group GN+M. The vector space Wλ(µ) = Hom G M (Wµ,Wλ) comes with a natural action of the group GN. Put n = λ1 − µ1 + λ2 − µ2 +.... In this article, for any standard Young tableau Ω of skew shape λ/µ we give a realization of Wλ(µ) as a subspace in the nfold tensor product (C N) ⊗ n, compatible with the action of the group GN. This subspace is determined as the image of a certain linear operator FΩ (M) on (C N) ⊗n, given by an explicit formula. When M = 0 and Wλ(µ) = Wλ is an irreducible representation of the group GN, we recover the classical realization of Wλ as a subspace in the space of all traceless tensors in (C N) ⊗n. Then the operator FΩ (0) may be regarded as the analogue for GN of the Young symmetrizer, corresponding to the standard tableau Ω of shape λ. This symmetrizer is a certain linear operator on (C N) ⊗n with the image equivalent to the irreducible polynomial representation of the complex general linear group GLN, corresponding to the partition λ. Even in the case M = 0, our formula for the operator FΩ (M) is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup GN of GLN. This twisted Yangian is a certain onesided coideal subalgebra of the Yangian corresponding to GLN. In particular, FΩ (M) is an intertwining operator between certain representations of the twisted Yangian in (C N) ⊗ n.
Casimir Operators and Monodromy Representations of Generalised Braid Groups
, 2003
"... Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group ..."
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Cited by 11 (6 self)
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Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group Bg of type g is a deformation of the action of (a finite extension of) W on V. The residues of ∇κ are the Casimirs κα of the subalgebras sl α 2 ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊆ V under the κα implies that, for generic values of the parameter, the braid group Bg acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g–modules if g = sl3 but that this is not the case if g ≇ sl2, sl3. We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin’s braid group Bn on the zero weight spaces of all simple U�sln–modules for n ≥ 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self–dual g–modules and obtain complete classification results for g = sln and g2 and conjecturally complete results for g orthogonal or symplectic.
Combinatorial representation theory
 in New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97), MSRI Publ. 38
, 1999
"... Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when ..."
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Cited by 11 (0 self)
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Abstract. We survey the field of combinatorial representation theory, describe the main results and main questions and give an update of its current status. Answers to the main questions are given in Part I for the fundamental structures, Sn and GL(n, �), and later for certain generalizations, when known. Background material and more specialized results are given in a series of appendices. We give a personal view of the field while remaining aware that there is much important and beautiful work that we have been unable to mention.
Irreducibility criterion for tensor products of Yangian evaluation modules
 Duke Math. J
"... Irreducibility criterion for tensor products of Yangian evaluation modules ..."
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Cited by 10 (1 self)
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Irreducibility criterion for tensor products of Yangian evaluation modules