Weyl Character Research Articles

Factorization of classical characters twisted by roots of unity: II

Fix natural numbers n⩾1, t⩾2 and a primitive tth root of unity ω. In previous work with A. Ayyer (2022) [5], we studied the factorization of specialized irreducible characters of GLtn, SO2tn+1, Sp2tn and O2tn evaluated at elements to ωjxi for 0⩽j⩽t−1 and 1⩽i⩽n. In this work, we extend the results to the groups GLtn+m(0⩽m⩽t−1), SO2tn+3, Sp2tn+2 and O2tn+2 evaluated at similar specializations: (1) for the GLtn+m case, we set the first tn elements to ωjxi for 0⩽j⩽t−1 and 1⩽i⩽n and the remaining m to y,ωy,…,ωm−1y; (2) for the other three families, the same specializations but with m=1. The main results of this paper are a characterization of partitions for which these characters vanish and a factorization of nonzero characters into those of smaller classical groups. Our motivation is the conjectures of Wagh and Prasad (2020) [18] relating the irreducible representations of Spin2n+1 and SL2n, SL2n+1 and Sp2n as well as Spin2n+2 and Sp2n. Our proofs use the Weyl character formulas and the beta-sets of t-core partitions. Lastly, we give a bijection to prove that there infinitely many t-core partitions for which these characters are nonzero.

$G$-equivariance of formal models of flag varieties

Let \mathbb{G} be a split connected reductive group scheme over the ring of integers \mathfrak{o} of a finite extension L|\mathbb{Q}_p and \lambda\in X(\mathbb{T}) an algebraic character of a split maximal torus \mathbb{T}\subseteq\mathbb{G} . Let us also consider the rigid analytic flag variety X^{\operatorname{rig}} of \mathbb{G} and G=\mathbb{G}(L) . In the first part of this paper, we introduce a family of \lambda -twisted differential operators on a formal model \mathfrak{Y} of X^{\operatorname{rig}} . We compute their global sections and we prove coherence together with several cohomological properties. In the second part, we define the category of coadmissible G -equivariant arithmetic \mathscr{D}(\lambda) -modules over the family of formal models of the rigid flag variety X^{\operatorname{rig}} . We show that if \lambda is such that \lambda + \rho is dominant and regular ( \rho being the Weyl character), then the preceding category is anti-equivalent to the category of admissible locally analytic G -representations, with central character \lambda . In particular, we generalize the main results from a paper by Huyghe, Patel, Schmidt and Strauch (2019) for algebraic characters.

WHITTAKER FUNCTIONS FROM MOTIVIC CHERN CLASSES

We prove a motivic analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized ag manifold G/B multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure–Lusztig (D–L) operators, identifies an Euler characteristic above to a formula of Brubaker, Bump and Licata for the Iwahori–Whittaker functions of the principal series representation of the p-adic Langlands dual group. As a corollary, we recover the classical Casselman–Shalika formula for the spherical Whittaker function. The proofs are based on localization in equivariant K-theory, and require a geometric interpretation of how the Hecke inverse of a D–L operator acts on the class of a point. We prove that the Hecke inverse operators give Grothendieck–Serre dual classes of the motivic classes, a result which might be of independent interest. In an Appendix jointly authored with Dave Anderson, we show that if the line bundle is trivial, we recover a generalization of a classical formula by Kostant, Macdonald, Shapiro and Steinberg for the Poincaré polynomial of G/B; the generalization we consider is due to Akyıldız and Carrell and replaces G/B by any smooth Schubert variety.

Demazure formula for A n Weyl polytope sums

The weights of finite-dimensional representations of simple Lie algebras are naturally associated with Weyl polytopes. Representation characters decompose into multiplicity-free sums over the weights in Weyl polytopes. The Brion formula for these Weyl polytope sums is remarkably similar to the Weyl character formula. Moreover, the same Lie characters are also expressible as Demazure character formulas. This motivates a search for new expressions for Weyl polytope sums, and we prove such a formula involving Demazure operators. It applies to the Weyl polytope sums of the simple Lie algebras An for all dominant integrable highest weights and all ranks n.

Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system

It is known that there is a correspondence between representations of superalgebras and ordinary (non-graded) algebras. Keeping in mind this type of correspondence between the twisted quantum affine superalgebra Uq(gl(2r|1)(2)) and the non-twisted quantum affine algebra Uq(so(2r+1)(1)), we proposed, in the previous paper [1], a Wronskian solution of the T-system for Uq(so(2r+1)(1)) as a reduction (folding) of the Wronskian solution for the non-twisted quantum affine superalgebra Uq(gl(2r|1)(1)). In this paper, we elaborate on this solution, and give a proof missing in [1]. In particular, we explain its connection to the Cherednik-Bazhanov-Reshetikhin (quantum Jacobi-Trudi) type determinant solution known in [2]. We also propose Wronskian-type expressions of T-functions (eigenvalues of transfer matrices) labeled by non-rectangular Young diagrams, which are quantum affine algebra analogues of the Weyl character formula for so(2r+1). We show that T-functions for spinorial representations of Uq(so(2r+1)(1)) are related to reductions of T-functions for asymptotic typical representations of Uq(gl(2r|1)(1)).

Character Polynomials and the Restriction Problem

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

Strange expectations and the Winnie-the-Pooh problem

Motivated by the study of simultaneous cores, we give three proofs (in varying levels of generality) that the expected norm of a weight in a highest weight representation Vλ of a complex simple Lie algebra g is 1h+1〈λ,λ+2ρ〉. First, we argue directly using the polynomial method and the Weyl character formula. Second, we relate this problem to the “Winnie-the-Pooh problem” regarding orthogonal decompositions of Lie algebras; although this approach offers the most explanatory power by interpreting the quantity 〈λ,λ+2ρ〉 as the eigenvalue for the Casimir element on Vλ, it applies only to Cartan types other than A and C. Third, we use the combinatorics of semistandard tableaux to obtain the result in type A. We conclude with computations of many combinatorial cumulants.

Importance of electron correlations in understanding photoelectron spectroscopy and Weyl character of MoTe2

We study the role of electron correlations in the presumed type-II Weyl semimetallic candidate $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{MoTe}}_{2}$ by employing density functional theory (DFT) where the on-site Coulomb repulsion (Hubbard $U)$ for the Mo $4d$ states is included within the DFT+$U$ scheme. We show that pure DFT calculations fail to describe important features of the light-polarization dependence of the angular resolved photoemission intensity which can be accounted for by including the role of the Hubbard $U$. At the same time, while pure DFT calculations cannot explain the angular dependence of the Fermi surface as revealed by quantum oscillation experiments (a fact which had raised doubt about the presence of the Weyl physics in $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{MoTe}}_{2})$, inclusion of such on-site Coulomb repulsion can. We find that while the number of Weyl points (WPs) and their position in the Brillouin zone change as a function of $U$, a pair of such WPs very close to the Fermi level survive the inclusion of these important corrections. Our calculations suggest that the Fermi surface of $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{MoTe}}_{2}$ is in the vicinity of a correlations-induced Lifshitz transition, which can be probed experimentally and its interplay with the Weyl physics might be intriguing.

Thermopower and Unconventional Nernst Effect in the Predicted Type-II Weyl Semimetal WTe2.

WTe2 is one of a series of recently discovered high mobility semimetals, some of whose properties are characteristic of topological Dirac or Weyl metals. One of its most interesting properties is the unsaturated giant magnetoresistance that it exhibits at low temperatures. An important question is the degree to which this property can be ascribed to a conventional semimetallic model in which a highly compensated, high mobility metal exhibits large magnetoresistance. Here, we show that the longitudinal thermopower (Seebeck effect) of semimetallic WTe2 exfoliated flakes exhibits periodic sign changes about zero with increasing magnetic field that indicates distinct electron and hole Landau levels and nearly fully compensated electron and hole carrier densities. However, inconsistent with a conventional semimetallic picture, we find a rapid enhancement of the Nernst effect at low temperatures that is nonlinear in magnetic field, which is consistent with Weyl points in proximity to the Fermi energy. Hence, we demonstrate the role played by the Weyl character of WTe2 in its transport properties.

Explicit Calculations of Tensor Product Coefficients for E7

We propose a new method to calculate coupling coefficients of E_7 tensor products. Our method is based on explicit use of E_7 characters in the definition of a tensor product. When applying Weyl character formula for E_7 Lie algebra, one needs to make sums over 2903040 elements of E_7 Weyl group. To implement such enormous sums, we show we have a way which makes their calculations possible. This will be accomplished by decomposing an E_7 character into 72 participating A_7 characters.

Non-Archimedean Whittaker Functions as Characters: A Probabilistic Approach to the Shintani–Casselman–Shalika Formula

For a reductive group $G$ over a non-Archimedean local field (e.g $GL_n( \mathbb{Q}_p )$ ), Jacquet's Whittaker function is essentially proportional to a character of an irreducible representation of the Langlands dual group $G^\vee( \mathbb{C} )$ ( a Schur function if $G = GL_n( \mathbb{Q}_p )$). We propose a probabilistic approach to this claim, known as the Shintani-Casselman-Shalika formula, when the group $G$ has at least one minuscule cocharacter in the coweight lattice. Our presentation goes along the following lines. Thanks to a minuscule random walk $W^{(z)}$ on the coweight lattice and a related random walk on the Borel subgroup, we establish a Poisson kernel formula for the non-Archimedean Whittaker function. The expression and its ingredients are similar to the one previously obtained by the author in the Archimedean case. A simple manipulation reduces the problem to evaluating the probability of $W^{(z)}$ never exiting the Weyl chamber. Then, an implementation of the reflection principle forces the appearance of the Weyl character formula and therefore retrieves characters of $G^\vee\left( \mathbb{C} \right)$. The construction of the random walk on the Borel subgroup requires some care. It is extracted from a spherical random walk whose increments have a distribution that can be understood as elements from the spherical Hecke algebra.

A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

Using a K-theory point of view, Bott related the Atiyah–Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.

Construction and deconstruction of single instanton Hilbert series

Many methods exist for the construction of the Hilbert series describing the moduli spaces of instantons. We explore some of the underlying group theoretic relationships between these various constructions, including those based on the Coulomb branches and Higgs branches of SUSY quiver gauge theories, as well as those based on generating functions derivable from the Weyl Character Formula. We show how the character description of the reduced single instanton moduli space of any Classical or Exceptional group can be deconstructed faithfully in terms of characters or modified Hall-Littlewood polynomials of its regular semi-simple subgroups. We derive and utilise Highest Weight Generating functions, both for the characters of Classical or Exceptional groups and for the Hall-Littlewood polynomials of unitary groups. We illustrate how the root space data encoded in extended Dynkin diagrams corresponds to relationships between the Coulomb branches of quiver gauge theories for instanton moduli spaces and those for T(SU(N)) moduli spaces.

Breaking classical Lie groups to finite subgroups – an automated approach

The decomposition of representations of compact classical Lie groups into representations of finite subgroups is discussed. A Mathematica package is presented that can be used to compute these branching rules using the Weyl character formula. For some low order finite groups including A4 and Δ(27) general analytical formulas are presented for the branching rules of arbitrary representations of their smallest Lie super-groups.

The six-vertex model and deformations of the Weyl character formula

We use statistical mechanics--variants of the six-vertex model in the plane studied by means of the Yang---Baxter equation--to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

A Generalization of Tokuyama's Formula to the Hall-Littlewood Polynomials

A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's parametrization for the Schur polynomials. We generalize Tokuyama's formula to the Hall-Littlewood polynomials by extending Tokuyama's statistics. Our result, in addition to specializing to Tokuyama's result and the aforementioned classical results, also yields connections to the monomial symmetric function and a new deformation of Stanley's formula.

Crystal graphs, Tokuyama’s theorem, and the Gindikin–Karpelevič formula for $$G_{2}$$ G 2

We conjecture a deformation of the Weyl character formula for type $$G_{2}$$G2 in the spirit of Tokuyama's formula for type $$A$$A. Using our conjecture, we prove a combinatorial version of the Gindikin---Karpelevia? formula for $$G_{2}$$G2, in the spirit of Bump---Nakasuji's formula for type $$A$$A.

Highest weight generating functions for Hilbert series

We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups. The method draws on plethystic functions and character generating functions along with Weyl integration. We give explicit examples showing how the use of such highest weight generating functions (HWGs) permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and also of the moduli spaces of instantons. We identify how the HWGs of gauge invariant operators of a selection of classical and exceptional SQCD theories result from the interaction under symmetrisation between a product group and the invariant tensors of its gauge group. In order to calculate HWGs, we derive and tabulate character generating functions for low rank groups by a variety of methods, including a general character generating function, based on the Weyl Character Formula, for any classical or exceptional group.

K-theory and the quantization commutes with reduction problem

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula.

The elliptic Weyl character formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].