Two positivity conjectures for Kerov polynomials

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the “free cumulants” of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov–Biane, recently proved by Féray.     

On Young's Orthogonal Form and the Characters of the Alternating Group

A combinatorial method of determining the characters of the alternating group is presented. We use matrix representations, due to Thrall, that are closely related to Young's orthogonal form of representations of the symmetric group. The characters are computed directly from matrix entries of these representations and entries of the character table of the symmetric group.     

The Bar Construction and Affine Stacks

We discuss the equivalence of two constructions of a unipotent group scheme attached to a differential graded algebra over a ?-algebra. The first construction is using the bar resolution and the second is using Toen's theory of affine stacks. We use this to establish the equivalence of two approaches to the comparison theorem in p-adic Hodge theory for the unipotent fundamental group of varieties defined over p-adic fields.     

Triangular Matrix Representations

In this paper we develop the theory of generalized triangular matrix representation in an abstract setting. This is accomplished by introducing the concept of a set of left triangulating idempotents. These idempotents determine a generalized triangular matrix representation for an algebra. The existence of a set of left triangulating idempotents does not depend on any specific conditions on the algebras; however, if the algebra satisfies a mild finiteness condition, then such a set can be refined to a “complete” set of left triangulating idempotents in which each “diagonal” subalgebra has no nontrivial generalized triangular matrix representation. We then apply our theory to obtain new results on generalized triangular matrix representations, including extensions of several well known results.     

Hook formula and related identities

A new proof of the hook formula for the dimension of representations of the symmetric group is given with the help of identities which are of independent interest. A probabilistic interpretation of the proof and new formulas relating the parameters of the Young diagrams are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 3–20, 1989.     

Filtration de certains espaces de fonctions sur un espace symétrique réductif

We introduce a filtration of a -module of some space of functions on a reductive symmetric space G/H, and compute the associated grading as a direct sum of induced representations. As an application of this result to the reductive groups viewed as symmetric spaces, we are able to realize any Harish-Chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups, the inducing representations being trivial on the unipotent radical.     

Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

We consider a specific class of coadjoint orbits of maximal unipotent subgroups in classical groups over a finite field; namely, orbits associated with orthogonal subsets in root systems. We derive a formula for the dimension of these orbits in terms of the Weyl group and construct polarizations for canonical forms on the orbits. As a consequence, we describe all possible dimensions of irreducible representations of such unipotent groups.     

Degenerate group of type A: Representations and flag varieties

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type A, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincaré-Birkhoff-Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties F _n ^a and their desingularizations R _n can be obtained via this construction. We prove that the coordinate ring of R _n is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type A.     

The Block Structure of the Images of Regular Unipotent Elements from Subsystem Symplectic Subgroups of Rank 2 in Irreducible Representations of Symplectic Groups. I

The dimensions of the Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type C2 in p-restricted irreducible representations of groups of type Cn in characteristic p ≥ 11 with locally small highest weights are found. These results can be applied for investigating the behavior of unipotent elements in modular representations of simple algebraic groups and recognizing representations and linear groups. The article consists of 3 parts. In the first one, preliminary lemmas that are necessary for proving the principal results, are contained and the case where all weights of the restriction of a representation considered to a subgroup of type A1 containing a relevant unipotent element are less than p, is investigated.     

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group’s relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.     

Poles of Intertwining Operators via Endoscopy; the Connection with Prehomogeneous Vector Spaces With an Appendix, `Basic Endoscopic Data', by Diana Shelstad To the Memory of Magdy Assem

In this paper, we determine the residues at poles of standard intertwining operators for parabolically induced representations of an arbitrary connected reductive quansisplit algebraic group over a p-acid field whenever the unipotent radical of the parabolic subgroup is Abelian. We then interpret these residues by means of the theory of endoscopy.     

Pieri and Littlewood–Richardson rules for two rows and cluster algebra structure

We study an algebra encoding a twice-iterated Pieri rule for the representations of the general linear group and prove that it has the structure of a cluster algebra. We also show that its cluster variables invariant under a unipotent subgroup generate the highest weight vectors of irreducible representations occurring in the decomposition of the tensor product of two irreducible representations of the general linear group one of whom is labeled by a Young diagram with less than or equal to two rows.     

Schur Orthogonality Relations for the Finitary Infinite Symmetric Group

Let S f be the finitary infinite symmetric group. For a certain class of irreducible unitary representations of S f , a version of Schur orthogonality relations is proved. That is, we construct an invariant inner product on the matrix coefficient space of each representation and show that matrix coefficients for distinct representations are orthogonal with respect to these norms.     

Schur Orthogonality Relations for the Finitary Infinite Symmetric Group

Let S f be the finitary infinite symmetric group. For a certain class of irreducible unitary representations of S f , a version of Schur orthogonality relations is proved. That is, we construct an invariant inner product on the matrix coefficient space of each representation and show that matrix coefficients for distinct representations are orthogonal with respect to these norms.     

Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications

Generalised matrix elements of the irreducible representations of the quantum SU(2) group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of sl(2). A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. Ser. A 66 (1990), 146–149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and new applications of this interpretation are presented.Supported by a NATO-Science Fellowship of the Netherlands Organization for Scientific Research (NWO).     

The Minimal Polynomials of Unipotent Elements in Irreducible Representations of the Special Linear Group

The minimal polynomials of images of unipotent elements in irreducible rational representations of a special linear group over an algebraically closed field of characteristic p > 2 are found. In particular, we show that the degree of such polynomial is equal to the order of an element provided the highest weight of a representation is in some sense large enough with respect to p.     

Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

An n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213–233) to have an α-symmetric distribution, α > 0, if its characteristic function is of the form φ(|ξ₁|^α + … + |ξ_n|^α). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous α-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on ⁿ. A new class of “zonally” symmetric stable laws on ⁿ is defined, and series expansions are derived for their characteristic functions and densities.     

On the smallest simple, unipotent Bol loop

Finite simple, unipotent Bol loops have recently been identified and constructed using group theory. However, the purely group-theoretical constructions of the actual loops are indirect, somewhat arbitrary in places, and rely on computer calculations to a certain extent. In the spirit of revisionism, this paper is intended to give a more explicit combinatorial specification of the smallest simple, unipotent Bol loop, making use of concepts from projective geometry and quasigroup theory along with the group-theoretical background. The loop has dual permutation representations on the projective line of order 5, with doubly stochastic action matrices.     

Generalized Whittaker vectors for holomorphic and quaternionic representations

The conjugacy class of parabolic subgroups with Heisenberg unipotent radical in a simple Lie groups over ³ not of type C_nC_{n} contains an element defined over  for each quaternionic real form. In this paper we study the Whittaker models for quaternionic discrete series of these real forms and prove results analogous and by analogous methods to the case of simple Lie groups over  that are the automorphism groups of tube type Hermitian symmetric domain and (so-called Bessel models) for holomorphic representations. In particular we calculate the decomposition of the space of Whittaker vectors under the action of the stabilizer of the corresponding character in a Levi factor of the Heisenberg parabolic subgroup.     

A subalgebra of 0-Hecke algebra

Let (W,I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.