Eric Marberg

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

Combinatorial methods of character enumeration for the unitriangular group

Abstract Let UT n ( q ) denote the group of unipotent n × n upper triangular matrices over a field with q elements. The degrees of the complex irreducible characters of UT n ( q ) are precisely the integers q e with 0 ⩽ e ⩽ ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ , and it has been conjectured that the number of irreducible characters of UT n ( q ) with degree q e is a polynomial in q − 1 with nonnegative integer coefficients (depending on n and e). We confirm this conjecture when e ⩽ 8 and n is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in n and q giving the number of irreducible characters of UT n ( q ) with degree q e when n > 2 e and e ⩽ 8 . When divided by q n − e − 2 and written in terms of the variables n − 2 e − 1 and q − 1 , these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of UT n ( q ) with degree ⩽ q 8 are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of UT n ( q ) .

Positivity conjectures for Kazhdan–Lusztig theory on twisted involutions: The finite case

Abstract Let ( W , S ) be any Coxeter system and let w ↦ w ⁎ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have shown that the corresponding set of twisted involutions (i.e., elements w ∈ W with w − 1 = w ⁎ ) naturally generates a module of the Hecke algebra of ( W , S ) with two distinguished bases. The transition matrix between these bases defines a family of polynomials P y , w σ which one can view as a “twisted” analogue of the much-studied family of Kazhdan–Lusztig polynomials of ( W , S ) . The polynomials P y , w σ can have negative coefficients, but display several conjectural positivity properties of interest, which parallel positivity properties of the Kazhdan–Lusztig polynomials. This paper reports on some calculations which verify four such positivity conjectures in several finite cases of interest, in particular for the non-crystallographic Coxeter systems of types H 3 and H 4 .

A supercharacter analogue for normality

Diaconis and Isaacs define in [8] (Diaconis and Isaacs, 2008) a supercharacter theory for algebra groups over a finite field by constructing certain unions of conjugacy classes called superclasses and certain reducible characters called supercharacters. This work investigates the properties of algebra subgroups H⊂G which are unions of some set of the superclasses of G; we call such subgroups supernormal. After giving a few useful equivalent formulations of this definition, we show that products of supernormal subgroups are supernormal and that all normal pattern subgroups are supernormal. We then classify the set of supernormal subgroups of Un(q), the group of unipotent upper triangular matrices over the finite field Fq, and provide a formula for the number of such subgroups when q is prime. Following this, we give supercharacter analogues for Cliffordʼs theorem and Mackeyʼs “method of little groups.” Specifically, we show that a supercharacter restricted to a supernormal subgroup decomposes as a sum of supercharacters with the same degree and multiplicity. We then describe how the supercharacters of an algebra group of the form Un=Uh⋉Ua, where Ua is supernormal and a2=0, are parametrized by Uh-orbits of the supercharacters of Ua and the supercharacters of the stabilizer subgroups of these orbits.

Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures

Abstract Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of “admissible sequences” by Richardson and Springer. They are maximal chains in Richardson and Springers weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of “involution Schubert polynomials” can be viewed as a Billey–Jockusch–Stanley type formula for cohomology class representatives of O n - and Sp 2 n -orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type B.

Schur P-positivity and involution Stanley symmetric functions

The involution Stanley symmetric functions

Automorphisms and generalized involution models of finite complex reflection groups

\hat{F}_y

Iterative character constructions for algebra groups

are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words, and are indexed by the involutions in the symmetric group. By construction each

Transition formulas for involution Schubert polynomials

\hat{F}_y

Strong forms of linearization for Hopf monoids in species

is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur