The t-analog of the level one string function for twisted affine Kac–Moody algebras

We study Lusztig?s t-analog of weight multiplicities, or affine Kostka–Foulkes polynomials, associated to level one representations of twisted affine Kac–Moody algebras. We obtain an explicit closed form expression for the unique t-string function, using constant term identities of Macdonald and Cherednik. This extends previous work on t-string functions for the untwisted simply-laced affine Kac–Moody algebras.     

An iterative formula for the Kostka–Foulkes polynomials

Journal of Algebraic Combinatorics - An iterative formula for the Kostka–Foulkes polynomials is given using the vertex operator realization of the Hall–Littlewood polynomials. The...     

Noncommutative symmetric functions and an amazing matrix

We present a simple way to derive the results of Diaconis and Fulman [P. Diaconis, J. Fulman, Foulkes characters, Eulerian idempotents, and an amazing matrix, arXiv:1102.5159] in terms of noncommutative symmetric functions.     

Branching formula for q-Littlewood–Richardson coefficients

A generalized inversion statistic is introduced on k-tuples of semistandard tableaux. It is shown that the cospin of a semistandard k-ribbon tableau is equal to the generalized inversion number of its k-quotient. This leads to a branching formula for the q-analogue of Littlewood–Richardson coefficients defined by Lascoux, Leclerc, and Thibon. This branching formula generalizes a recurrence of Garsia and Procesi involving Kostka–Foulkes polynomials.     

On the geometry and quantization of symplectic Howe pairs

We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical correspondence between the orbit spaces of the respective moment images. Furthermore, we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group. In the Kähler case we show that the linear representation of a pair of compact connected Lie groups on the geometric quantization of the manifold is then equipped with a representation-theoretic Howe duality.     

Classifying patterns by automorphism group: An operator theoretic approach

This paper considers the problem of enumerating the elements of a set under a group action with a given automorphism group. The problem is approached from a linear algebraic point of view, with a class of identities obtained by applications of appropriate linear operators and functionals. A variety of new counting and enumerating results are obtained in this manner, and the connections to the recent work of de Bruijn, Foulkes, Sheehan, Stockmeyer and White are defined. Included among the new results are general formulas for enumerating patterns with a given automorphism group when a group acts on the range and domain of a finite function space. In this case, the multilinear computing techniques developed by Williamson are exploited.     

On Doi–Naganuma and Shimura liftings

The Ramanujan Journal - In this paper, we extend the Doi–Naganuma lifting to higher levels by following the methods of Zagier and Kohnen. We prove that there is a Hecke-equivariant linear map...     

On differential operators over a map,thick morphisms of supermanifolds,and symplectic micromorphisms

We recall the notion of a differential operator over a map (in linear and non-linear settings) and consider its versions such as formal ħ-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and operators arising as quantization of symplectic micromorphisms.     

Foulkes characters, Eulerian idempotents, and an amazing matrix

John Holte (Am. Math. Mon. 104:138?C149, 1997) introduced a family of ??amazing matrices?? which give the transition probabilities of ??carries?? when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545?C556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.     

Jacobi-like forms,differential equations,and Hecke operators

We construct a map from the space of Jacobi-like forms [image omitted]() for a discrete subgroup [image omitted] to the space [image omitted] of sequences of meromorphic functions satisfying certain conditions determined by some linear ordinary differential operators and prove that the Hecke operator actions on [image omitted]() and on [image omitted] are compatible with respect to this map.