A combinatorial rule for (co)minuscule Schubert calculus

We prove a root system uniform, concise combinatorial rule for Schubert calculus of minuscule and cominuscule flag manifolds G/P (the latter are also known as compact Hermitian symmetric spaces). We connect this geometry to the poset combinatorics of Proctor, thereby giving a generalization of Schützenberger's jeu de taquin formulation of the Littlewood-Richardson rule that computes the intersection numbers of Grassmannian Schubert varieties. Our proof introduces cominuscule recursions, a general technique to relate the numbers for different Lie types.     

A symplectic jeu de taquin bijection between the tableaux of King and of De Concini

The definitions, methods, and results are entirely combinatorial. The symplectic jeu de taquin algorithm developed here is an extension of Schützenberger's original jeu de taquin and acts on a skew form of De Concini's symplectic standard tableaux. This algorithm is used to construct a weight preserving bijection between the two most widely known sets of symplectic tableaux. Anticipated applications to Knuth relations and to decomposing symplectic tensor products are indicated.

     

From Macdonald polynomials to a charge statistic beyond type A

The charge is an intricate statistic on words, due to Lascoux and Schützenberger, which gives positive combinatorial formulas for Lusztig?s t-analogue of weight multiplicities and the energy function on affine crystals, both of type A. As these concepts are defined for all Lie types, it has been a long-standing problem to express them based on a generalization of charge. I present a method for addressing this problem in classical Lie types, based on the recent Ram-Yip formula for Macdonald polynomials and the quantum Bruhat order on the corresponding Weyl group. The details of the method are carried out in type A (where we recover the classical charge) and type C (where we define a new statistic).     

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

Compression of Nakajima monomials in type A and C

We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Kashiwara–Nakashima tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we define an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.     

On combinatorial formulas for Macdonald polynomials

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms.     

A combinatorial formula for Macdonald polynomials

In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund, Haiman and Loehr for Macdonald polynomials of type GLn. At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent and Littelmann).     

Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S₃-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.     

Combinatorial Models for Weyl Characters

We present a set of axioms for combinatorial objects closely related to those for Kashiwara's crystals. We show that any model for the axioms, such as Littelmann's path model, has a character—a nonnegative sum of irreducible characters for a semisimple Lie group or algebra, or more generally, a symmetrizable Kac-Moody algebra. Moreover, there are simple explicit restriction rules and rules for decomposing the product of any such character by an irreducible character.     

Rational tableaux and the tensor algebra of gln

A generalization of the usual column-strict tableaux (equivalent to a construction of R. C. King) is presented as a natural combinatorial tool for the study of finite dimensional representations of GL_n(C). These objects are called rational tableaux since they play the same role for rational representations of GL_n as ordinary tableaux do for polynomial representations. A generalization of Schensted's insertion algorithm is given for rational tableaux, and is used to count the. multiplicities of the irreducible GL_n-modules in the tensor algebra of GL_n. The problem of counting multiplicities when the kth tensor power gl_n^⊗k is decomposed into modules which are simultaneously irreducible with respect to GL_n and the symmetric group S_k is also considered. The existence of an insertion algorithm which describes this decomposition is proved. A generalization of border strip tableaux, in which both addition and deletion of border strips is allowed, is used to describe the characters associated with this decomposition. For large n, these generalized border strip tableaux have a simple structure which allows derivation of identities due to Hanlon and Stanley involving the (large n) decomposition of gl_n^⊗k.     

On the Orthogonal Tableaux of Koike and Terada

Many different definitions have been given for semistandard odd and even orthogonal tableaux which enumerate the corresponding irreducible orthogonal characters. Weightpreserving bijections have been provided between some of these sets of tableaux (see [3], [8]). We give bijections between the (semistandard) odd orthogonal Koike-Terada tableaux and the odd orthogonal Sundaram-tableaux and between the even orthogonal Koike-Terada tableaux and the even orthogonal King-Welsh tableaux. As well, we define an even version of orthogonal Sundaram tableaux which enumerate the irreducible characters of O(2n).     

Signature Characters of Highest-Weight Representations of $U_{q}(\mathfrak {gl}_{n})$

We consider \(U_{q}(\mathfrak {gl}_{n})\), the quantum group of type A for |q| = 1, q generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the technique involves combinatorics of the Gelfand-Tsetlin bases. As an application, we obtain information about unitarity of finite-dimensional irreducible representations for arbitrary q: we classify the continuous spectrum of the unitarity locus. We also recover some known results in the classical limit \(q \rightarrow 1\) that were obtained by different means. Finally, we provide several explicit examples of signature characters.     

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