Representation-theoretic Proof Research Articles

On a family of Siegel Poincaré series

Let [Formula: see text] be a congruence subgroup of [Formula: see text]. Using Poincaré series of [Formula: see text]-finite matrix coefficients of integrable discrete series representations of [Formula: see text], we construct a spanning set for the space [Formula: see text] of Siegel cusp forms of weight [Formula: see text]. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of [Formula: see text], we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of [Formula: see text]. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.

Polynomial induction and the restriction problem

We construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of a general linear group to the category of representations of its Weyl group. This construction leads to a representation-theoretic proof of Littlewood's plethystic formula for the multiplicity of an irreducible representation of the symmetric group in such a restriction. The unimodality of certain bipartite partition functions follows.

Boolean Product Polynomials, Schur Positivity, and Chern Plethysm

Abstract Let $k \leq n$ be positive integers, and let $X_n = (x_1, \dots , x_n)$ be a list of $n$ variables. The Boolean product polynomial $B_{n,k}(X_n)$ is the product of the linear forms $\sum _{i \in S} x_i$, where $S$ ranges over all $k$-element subsets of $\{1, 2, \dots , n\}$. We prove that Boolean product polynomials are Schur positive. We do this via a new method of proving Schur positivity using vector bundles and a symmetric function operation we call Chern plethysm. This gives a geometric method for producing a vast array of Schur positive polynomials whose Schur positivity lacks (at present) a combinatorial or representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for certain $k$ to other combinatorial objects including derangements, positroids, alternating sign matrices, and reverse flagged fillings of a partition shape. We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group ${\mathfrak{S}}_n$ on a divergence free quotient of superspace.

Quantum information processing and composite quantum fields

Some beautiful identities involving hook contents of Young diagrams have been found in the field of quantum information processing, along with a combinatorial proof. We here give a representation theoretic proof of these identities and a number of generalizations. Our proof is based on trace identities for elements belonging to a class of permutation centralizer algebras. These algebras have been found to underlie the combinatorics of composite gauge invariant operators in quantum field theory, with applications in the AdS/CFT correspondence. Based on these algebras, we discuss some analogies between quantum information processing tasks and the combinatorics of composite quantum fields and argue that this can be fruitful interface between quantum information and quantum field theory, with implications for AdS/CFT.

The HOMFLY polynomial of links in closed braid form

We provide a new proof of Jaeger’s formula expressing the HOMFLY polynomial of a link presented in closed braid form, replacing the original representation theoretic proof with an easy combinatorial and geometric argument. Using new variants of Jaeger’s result we provide a direct and elementary proof of the fact that the braid index of a link that has an n-string closed braid diagram that is also reduced and alternating, is exactly n. Until now this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.

Multiplication Formulas and Canonical Bases for Quantum Affine gln

AbstractWe will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra of a cyclic quiver Δ(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group .

English

We consider separately radial (with corresponding group Tn) and radial (with corresponding group U(n)) symbols on the projective space Pn(C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between Tn and U(n).

Pieri Rules for Classical Groups and Equinumeration between Generalized Oscillating Tableaux and Semistandard Tableaux

We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretic proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.

The conformal four-point integrals, magic identities and representations of U(2,2)

In [3,5] we found mathematical interpretations of the one-loop conformal four-point Feynman integral as well as the vacuum polarization Feynman integral in the context of representations of a Lie group U(2,2) and quaternionic analysis. Then we raised a natural question of finding mathematical interpretation of other Feynman diagrams in the same setting. In this article we describe this interpretation for all conformal four-point integrals. Using this interpretation, we give a representation-theoretic proof of an operator version of the “magic identities” for the conformal four-point integrals described by the box diagrams.The original “magic identities” are due to J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev; they assert that all n-loop box integrals for four scalar massless particles are equal to each other [2]. The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true for the Minkowski metric case. However, the Minkowski case is much more subtle. In this article we prove an operator version of the magic identities in the Minkowski metric case and, in particular, specify the relative positions of cycles of integration that make these identities correct.No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

A representation-theoretic proof of the branching rule for Macdonald polynomials

We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of U_q(gl_n). In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture.

A representation-theoretic proof of the branching rule for Macdonald polynomials

We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof-Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of $U_q(gl_n)$. In the Gelfand-Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald's operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl-Kasatani conjecture. Nous donnons une nouvelle preuve représentation-théorique de la règle de branchement pour les polynômes de Macdonald en utilisant l’expression Etingof-Kirillov Jr. pour les polynômes de Macdonald comme des traces de intertwiners de $U_q(gl_n)$. Dans la base de Gelfand-Tsetlin, nous montrons que les éléments de matrice diagonaux de ces intertwiners sont donnés par action des opérateurs de Macdonald à un noyau simple. Un ingrédient essentiel dans la preuve est une application entre les parties sphériques des algèbres de Hecke double affines de rangs différents basés sur la conjecture Dunkl-Kasatani.

Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces

Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ ·‖) is a p-convex Banach space then for any Lipschitz function f : H→ X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x)− f(y)‖ dW (x, y) . ( log log dW (x, y) log dW (x, y) )1/p . (1) We also show that any embedding into X of a ball of radius R > 4 in H incurs bi-Lipschitz distortion that grows at least as a constant multiple of ( logR log logR )1/p . (2) Both (1) and (2) are sharp up to the iterated logarithm terms. When X is Hilbert space we obtain a representation-theoretic proof yielding bounds corresponding to (1) and (2) which are sharp up to a universal constant.

A Mirror Symmetric Solution to the Quantum Toda Lattice

We give a representation-theoretic proof of a conjecture from Rietsch (Adv Math 217:2401–2442, 2008) providing integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SLn/B in a uniform way to arbitrary type, and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive and totally negative critical point of the ‘superpotential’ in every mirror fiber.

Integral formula of the unitary inversion operator for the minimal representation of $O(p, q)$

The indefinite orthogonal group $G = O(p,q)$ has a distinguished infinite dimensional unitary representation $\pi$, called the \textit{minimal representation} for $p+q$ even and greater than $6$. The \textit{Schr\"odinger model} realizes $\pi$ on a very simple Hilbert space, namely, $L^2(C)$ consisting of square integrable functions on a Lagrangean submanifold $C$ of the minimal nilpotent coadjoint orbit, whereas the $G$-action on $L^2(C)$ has not been well-understood. This paper gives an explicit formula of the unitary operator $\pi(w_0)$ on $L^2(C)$ for the `conformal inversion' $w_0$ as an integro-differential operator, whose kernel function is given by a Bessel distribution. Our main theorem generalizes the classic Schr\"odinger model on $L^2(\mathbf R^n)$ of the Weil representation, and leads us to an explicit formula of the action of the whole group $O(p,q)$ on $L^2(C)$. As its corollaries, we also find a representation theoretic proof of the inversion formula and the Plancherel formula for Meijer's $G$-transforms.

Spaces of Lattice Diagram Polynomials in One Set of Variables

For a partition μ of n, let Mμ be the space span of all partial derivatives of the alternate associated to μ in two n-sets of variables X and Y. The n!-theorem of [M. Haiman, J. Amer. Math. Soc.14 (2001), 941–1006] states that the dimension of Mμ is n!. In [F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Adv. Math.142 (1999), 244–334], we introduced slightly more general spaces Mμ/ij as a tool for an elementary proof of the n!-theorem. For this one needs a formula for the bigraded characters in terms of Macdonald polynomials that would follow from a representation theoretic proof of our four-term recurrence. Here we give an explicit basis for the Y-free component of the space Mμ/ij, and using this description we prove the Y-free analog of the four-term recurrence. Our basis of Mμ/ij has the further nice feature that it is a natural generalization of the Artin basis for the space of harmonic (covariant) polynomials for the symmetric group.

Geometrical Problems (1996)/Diane E. Tesler (b. 1944)

For a partition μ of n, let Mμ be the space span of all partial derivatives of the alternate associated to μ in two n-sets of variables X and Y. The n!-theorem of [M. Haiman, J. Amer. Math. Soc.14 (2001), 941–1006] states that the dimension of Mμ is n!. In [F. Bergeron, N. Bergeron, A. Garsia, M. Haiman, and G. Tesler, Adv. Math.142 (1999), 244–334], we introduced slightly more general spaces Mμ/ij as a tool for an elementary proof of the n!-theorem. For this one needs a formula for the bigraded characters in terms of Macdonald polynomials that would follow from a representation theoretic proof of our four-term recurrence. Here we give an explicit basis for the Y-free component of the space Mμ/ij, and using this description we prove the Y-free analog of the four-term recurrence. Our basis of Mμ/ij has the further nice feature that it is a natural generalization of the Artin basis for the space of harmonic (covariant) polynomials for the symmetric group.

A Brauer Algebra Theoretic Proof of Littlewood's Restriction Rules

LetUbe a complex vector space endowed with an orthogonal or symplectic form, and letGbe the subgroup ofGL(U) of all the symmetrics of this form (resp.O(U) orSp(U)); ifMis an irreducibleGL(U)-module, the Littlewood's restriction rule describes theG-moduleM|GL(U)G. In this paper we give a new representation-theoretic proof of this formula: realizingMin a tensor powerU⊗fand using Schur's duality, we reduce to the problem of describing the restriction to an irreducibleSf-module of an irreducible module for the centralizer algebra of the action ofGonU⊗f; the latter is a quotient of the Brauer algebra, and we know the kernel of the natural epimorphism, whence we deduce the Littlewood's restriction rule.

Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory

Capelli's identity plays a prominent role in Weyl's approach to Classical Invariant Theory. Capelli's identity was recently considered by Howe and Howe and Umeda. Howe gave an insightful representation-theoretic proof of Capelli's identity, and a similar approach was used to prove Turnbull's symmetric analog, as well as a new anti-symmetric analog, that was discovered independently by Kostant and Sahi. The Capelli, Turnbulll, and Howe-Umeda-Kostant-Sahi identities immediately imply, and were inspired by, identities of Cayley, Garding, and Shimura, respectively. In this paper, we give short combinatorial proofs of Capelli's and Turnbull's identities, and raise the hope that someone else will use our approach to prove the new Howe-Umeda-Kostant-Sahi identity.

Representation theoretic realizations of two classical character formulas of d.e.littlewood

In characteristic zero we construct an exact sequence over the symplectic group whose Euler-Poincare characteristic gives the branching rule for this group.This exact sequence is one way of realizing the Giambelli formula for the symplectic group.Also we give a representation-theoretic proof of the restriction rule for the symplectic group.