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Mark Shimozono

Publications and source records attributed to Mark Shimozono.

At least 37 records · Page 2Linked to original sources
arXiv2016

A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X

We establish the equality of the specialization $P_λ(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_λ(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine Lie algebras. This is achieved by constructing two uniform combinatorial models for the crystals associated with the mentioned tensor products: the quantum alcove model (which is naturally associated to Macdonald polynomials), and the quantum Lakshmibai-Seshadri path model. We provide an explicit affine crystal isomorphism between the two models, and realize the energy function in both models. In particular, this gives the first proof of the positivity of the $t = 0$ limit of the symmetric Macdonald polynomial in the untwisted and non-simply-laced cases, when it is expressed as a linear combination of the irreducible characters for a finite-dimensional simple Lie subalgebra, as well as a representation-theoretic meaning of the coefficients in this expression in terms of degree functions.

math.QA↗
arXiv2015

Explicit description of the degree function in terms of quantum Lakshmibai-Seshadri paths

We give an explicit and computable description, in terms of the parabolic quantum Bruhat graph, of the degree function defined for quantum Lakshmibai-Seshadri paths, or equivalently, for "projected" (affine) level-zero Lakshmibai-Seshadri paths. This, in turn, gives an explicit and computable description of the global energy function on tensor products of Kirillov-Reshetikhin crystals of one-column type, and also of (classically restricted) one-dimensional sums.

math.QA↗
arXiv2014

Virtual crystals and Kleber's algorithm

Kirillov and Reshetikhin conjectured what is now known as the fermionic formula for the decomposition of tensor products of certain finite dimensional modules over quantum affine algebras. This formula can also be extended to the case of $q$-deformations of tensor product multiplicities as recently conjectured by Hatayama et al. (math.QA/9812022 and math.QA/0102113). In its original formulation it is difficult to compute the fermionic formula efficiently. Kleber (q-alg/9611032 and math.QA/9809087) found an algorithm for the simply-laced algebras which overcomes this problem. We present a method which reduces all other cases to the simply-laced case using embeddings of affine algebras. This is the fermionic analogue of the virtual crystal construction by the authors, which is the realization of crystal graphs for arbitrary quantum affine algebras in terms of those of simply-laced type.

math.QA↗
arXiv2014

Quantum Lakshmibai-Seshadri paths and root operators

We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path.

math.QA↗
arXiv2013

A uniform model for Kirillov-Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph

We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a second paper to establish a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals, and the equality, for untwisted affine root systems, between the Macdonald polynomial with t set to zero and the graded character of tensor products of one-column KR modules.

math.QA↗
arXiv2013

Specializations of nonsymmetric Macdonald-Koornwinder polynomials

This work records the details of the Ram-Yip formula for nonsymmetric Macdonald-Koornwinder polynomials for the double affine Hecke algebras of not-necessarily-reduced affine root systems. It is shown that the t=0 equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type the proof requires the Ram-Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given for the limit at t equal to infinity. As a consequence we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at q equal to infinity, which yields the p-adic Iwahori-Whittaker functions of Brubaker, Bump, and Licata.

math.QA↗
arXiv2013

$k$-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

math.CO↗
arXiv2012

A uniform model for Kirillov-Reshetikhin crystals. Extended abstract

We present a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai-Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at t=0 and the graded character of a tensor product of KR modules.

math.RT↗
arXiv2012

Equivariant K-Chevalley Rules for Kac-Moody Flag Manifolds

Explicit combinatorial cancellation-free rules are given for the product of an equivariant line bundle class with a Schubert class in the torus-equivariant K-theory of a Kac-Moody flag manifold. The weight of the line bundle may be dominant or antidominant, and the coefficients may be described either by Lakshmibai-Seshadri paths or by the alcove model of the first author and Postnikov. For Lakshmibai-Seshadri paths, our formulas are the Kac-Moody generalizations of results of Griffeth-Ram and Pittie-Ram for finite dimensional flag manifolds. A gap in the proofs of the mentioned results is addressed.

math.CO↗
arXiv2011

From double quantum Schubert polynomials to k-double Schur functions via the Toda lattice

We show that the k-double Schur functions defined by the authors, and the quantum double Schubert polynomials studied by Kirillov and Maeno and by Ciocan-Fontanine and Fulton, can be obtained from each other by an explicit rational substitution. The main new ingredient is an explicit computation of Kostant's solution to the Toda lattice in terms of equivariant Schubert classes.

math.CO↗
arXiv2011

Quantum double Schubert polynomials represent Schubert classes

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define parabolic analogues of quantum double Schubert polynomials, and show that they represent Schubert classes in the equivariant quantum cohomology of partial flag varieties. For complete flags Anderson and Chen have announced a proof with different methods.

math.CO↗
arXiv2011

Equivariant Pieri Rule for the homology of the affine Grassmannian

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL_n and a similar formula is conjectured for Sp_{2n} and SO_{2n+1}. For SL_n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL_n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.

math.CO↗
arXiv2011

Affine crystals, one-dimensional sums and parabolic Lusztig q-analogues

This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of the third author and Zabrocki by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of the root system considered is sufficiently large. As a consequence, any one-dimensional sum associated to a classical affine root system with sufficiently large rank can be regarded as a parabolic Lusztig q-analogue.

math.QA↗
arXiv2011

X = K under review

We review the X = K conjecture and important ingredients for the proof. We also attach notes on the rank estimate for the X = K theorem to hold and on the strange relation that was found to be valid without the assumption that the rank is sufficiently large. Using the latter one obtains an algorithm to calculate the image of the combinatorial R-matrix and the value of the coenergy function.

math.QA↗
arXiv2010

k-shape poset and branching of k-Schur functions

We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is achieved by studying the combinatorics of a new class of partitions called k-shapes, which interpolates between k-cores and k+1-cores. We define a symmetric function for each k-shape, and show that they expand positively in terms of dual k-Schur functions. We obtain an explicit combinatorial description of the expansion of an ungraded k-Schur function into k+1-Schur functions. As a corollary, we give a formula for the Schur expansion of an ungraded k-Schur function.

math.CO↗
arXiv2009

K-theory Schubert calculus of the affine Grassmannian

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case G = SL_n, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, called K-k-Schur functions, whose highest degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations using Kashiwara's thick affine flag manifold.

math.CO↗