Supercharacters and superclasses for algebra groups

We study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups. These groups generalize the unimodular upper triangular groups over a finite field, and the supercharacter theory we develop extends results of Carlos André and Ning Yan that were originally proved in the upper triangular case. This theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table. We discuss connections with the Kirillov orbit method and with Gelfand pairs, and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class, respectively. We also show that products of supercharacters are positive integer combinations of supercharacters.

     

Heisenberg characters, unitriangular groups, and Fibonacci numbers

Let Un(Fq) denote the group of unipotent n×n upper triangular matrices over a finite field with q elements. We show that the Heisenberg characters of Un+1(Fq) are indexed by lattice paths from the origin to the line x+y=n using the steps (1,0), (1,1), (0,1), (0,2), which are labeled in a certain way by nonzero elements of Fq. In particular, we prove for n?1 that the number of Heisenberg characters of Un+1(Fq) is a polynomial in q−1 with nonnegative integer coefficients and degree n, whose leading coefficient is the nth Fibonacci number. Similarly, we find that the number of Heisenberg supercharacters of Un(Fq) is a polynomial in q−1 whose coefficients are Delannoy numbers and whose values give a q-analogue for the Pell numbers. By counting the fixed points of the action of a certain group of linear characters, we prove that the numbers of supercharacters, irreducible supercharacters, Heisenberg supercharacters, and Heisenberg characters of the subgroup of Un(Fq) consisting of matrices whose superdiagonal entries sum to zero are likewise all polynomials in q−1 with nonnegative integer coefficients.     

Constituents of supercharacters and Kirillov functions

Supercharacters and Kirillov functions have been proposed as two related constructions that mimic the characters of certain p-groups. The irreducible constituents of these functions are intimately linked; specifically, irreducible constituents of a Kirillov function are constituents of the corresponding supercharacter.     

Stable rigged configurations for quantum affine algebras of nonexceptional types

For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with Littlewood–Richardson coefficients. Combining this result with Kirillov et al. (2002) [13] and Lecouvey et al. (2011) [18] we settle the X=M conjecture under the large rank hypothesis.     

Generalized complex and Dirac structures on homogeneous spaces

The aim of this paper is to study generalized complex geometry (Hitchin, 2002) [6] and Dirac geometry (Courant, 1990) [3], (Courant and Weinstein, 1988) [4] on homogeneous spaces. We offer a characterization of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in sln(R). For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subalgebras.     

Non-commutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.     

Character Formulas for q-Rook Monoid Algebras

The q-rook monoid R _n(q) is a semisimple (q)-algebra that specializes when q 1 to [R _n], where R _n is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. We use a Schur-Weyl duality between R _n(q) and the quantum general linear group to compute a Frobenius formula, in the ring of symmetric functions, for the irreducible characters of R _n(q). We then derive a recursive Murnaghan-Nakayama rule for these characters, and we use Robinson-Schensted-Knuth insertion to derive a Roichman rule for these characters. We also define a class of standard elements on which it is sufficient to compute characters. The results for R _n(q) specialize when q = 1 to analogous results for R _n.     

Wall crossings for double Hurwitz numbers

Double Hurwitz numbers count covers of P¹ by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification (Goulden et al., 2005) [10], and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0 (Shadrin et al., 2008) [15]. This paper gives a unified approach to these results and strengthens them in several ways — the most important being the extension of the results of Shadrin et al. (2008) [15] to arbitrary genus.The main tool is the authors? previous work (Cavalieri et al., 2010) [6] expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987) [17], and could have broader applications.     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

The ubiquity of generalized cluster categories

Associated with a finite-dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in Amiot (2009) [1]. It was shown to be triangulated, and 2-Calabi–Yau when it is Hom-finite. By definition, the cluster categories of Buan et al. (2006) [4] are a special case. In this paper we show that a large class of 2-Calabi–Yau triangulated categories, including those associated with elements in Coxeter groups from Buan et al. (2009) [7], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in Amiot (2009) [1].     

Moment vanishing problem and positivity: Some examples

We consider the problem of vanishing of the momentswith Ω a compact domain in Rn and P(x), q(x) complex polynomials in x∈Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34] and [35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture:

Conjecture A. 

For positive μ ifmk(P,1)=0fork=1,2,… , thenmk(P,q)=0fork?1for any q.     

Representation theory of p-adic groups and canonical bases

In this paper, we interpret the Gindikin–Karpelevich formula and the Casselman–Shalika formula as sums over Kashiwara–Lusztig?s canonical bases, generalizing the results of Bump and Nakasuji (2010) [7] to arbitrary split reductive groups. We also rewrite formulas for spherical vectors and zonal spherical functions in terms of canonical bases. In a subsequent paper Kim and Lee (preprint) [14], we will generalize these formulas to p-adic affine Kac–Moody groups.     

Fischer Matrices of the Affine Groups

Let GL_n(q) be the general linear group and let H_n ; V_n(q) · GL_n(q) denote the affine group of V_n(q). In [1] and [4], we determined Fischer matrices for the conjugacy classes of GL_n(q) where n = 2, 3, 4 and we obtained the number of conjugacy classes and irreducible characters of H₂, H₃, and H₄. In this paper, we find the Fischer matrices of the affine group H_n for arbitrary n.AMS Subject Classification Primary 20C15 Secondary 20C33     

Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as where j is either 0 or 1. If j=0 then d?5 is an odd integer and n is an even integer satisfying 2?n?(d+1)/2. If j=1 then d?3 is an integer and n is an integer with converse parity with d and satisfying 0<n?[(d+1)/3] where [⋅] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations.     

Approximation of RNA multiple structural alignment

In the context of non-coding RNA (ncRNA) multiple structural alignment, Davydov and Batzoglou (2006) introduced in [7] the problem of finding the largest nested linear graph that occurs in a set G of linear graphs, the so-called Max-NLS problem. This problem generalizes both the longest common subsequence problem and the maximum common homeomorphic subtree problem for rooted ordered trees.In the present paper, we give a fast algorithm for finding the largest nested linear subgraph of a linear graph and a polynomial-time algorithm for a fixed number (k) of linear graphs. Also, we strongly strengthen the result of Davydov and Batzoglou (2006) [7] by proving that the problem is NP-complete even if G is composed of nested linear graphs of height at most 2, thereby precisely defining the borderline between tractable and intractable instances of the problem. Of particular importance, we improve the result of Davydov and Batzoglou (2006) [7] by showing that the Max-NLS problem is approximable within ratio in O(kn²) running time, where mopt is the size of an optimal solution. We also present O(1)-approximation of Max-NLS problem running in O(kn) time for restricted linear graphs. In particular, for ncRNA derived linear graphs, a -approximation is presented.     

Characters of modular torsion free representations of classical lie algebras

In this paper, we analyze the characters of modular, irreducible rep-resentations of classical Lie algebras g of types A_l-1 and Ci arising from a characteristic 0 construction of torsion free representations. By character, we refer to linear functionals on g identified with algebra homomorphisms from a distinguished central subalgebra O of the universal enveloping algebra of g. If Lie(G') = g, then for each character X standard representatives with respect to a fixed toral subalgebra are found in the (2-orbit containing the character X For many parameters, these characters are nilpotent. Furthermore, modular representations of type A_l-1 and type C_l Lie algebras constructed by induction from these irreducible, torsion free representations are shown to admit characters in a family of both Richardson and non-Richardson nilpotent orbits. Through this explicit induction construction, irreducible representations of minimal p-power dimension under the Kac-Weisfeiler conjecture are realized