Lakshmibai Seshadri Paths Research Articles

Seshadri stratifications and standard monomial theory

We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. We show that the Seshadri stratification provides a geometric setup for a standard monomial theory. In this framework, Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation as successive vanishing orders of regular functions.

Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application, we give new tableau models for the crystal bases of a level-zero fundamental representation and a level-zero extremal weight module over a quantum affine algebra of classical untwisted type, which we call quantum Kashiwara-Nakashima columns and semi-infinite Kashiwara-Nakashima tableaux. We give an explicit description of the crystal isomorphisms among three different realizations of the crystal basis of a level-zero fundamental representation by quantum Lakshmibai-Seshadri paths, quantum Kashiwara-Nakashima columns, and (ordinary) Kashiwara-Nakashima columns.

Crystals of Lakshmibai-Seshadri paths and extremal weight modules over quantum hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by the crystal of all Lakshmibai-Seshadri paths of shape λ. Let be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector of weight λ, and let be the crystal basis of with the element corresponding to We prove that the connected component of containing is isomorphic, as a crystal, to the connected component of containing the straight line Furthermore, we prove that if λ satisfies a special condition, then the crystal basis is isomorphic, as a crystal, to the crystal As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in where are the fundamental weights, in the case that is symmetric.

Chevalley formula for anti-dominant weights in the equivariant K-theory of semi-infinite flag manifolds

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17], the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17], our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

A study of Kostant-Kumar modules via Littelmann paths

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be of independent interest. We also discuss the existence of PRV components and generalised PRV components in KK modules.

Equivariant K -theory of semi-infinite flag manifolds and the Pieri–Chevalley formula

We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $\mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $\mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $\mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $\mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.

Path model for an extremal weight module over the quantized hyperbolic Kac-Moody algebra of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2, and set where are the fundamental weights. Denote by the extremal weight module of extremal weight λ with the extremal weight vector, and by the crystal basis of with the element corresponding to We prove that (i) is connected, (ii) the subset of elements of weight μ in is a finite set for every integral weight μ, and (iii) every extremal element in is contained in the Weyl group orbit of (iv) is irreducible. Finally, we prove that the crystal basis is isomorphic, as a crystal, to the crystal of Lakshmibai-Seshadri paths of shape λ.

Tensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths

Let λ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let QLS(λ) denote the set of quantum Lakshmibai-Seshadri (QLS) paths of shape λ. For an element w of a finite Weyl group W, the specializations at t=0 and t=∞ of the nonsymmetric Macdonald polynomial Ewλ(q,t) are explicitly described in terms of QLS paths of shape λ and the degree function defined on them. Also, for (level-zero) dominant integral weights λ, μ, we have an isomorphism Θ:QLS(λ+μ)→QLS(λ)⊗QLS(μ) of crystals. In this paper, we study the behavior of the degree function under the isomorphism Θ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.

Lakshmibai–Seshadri paths for hyperbolic Kac–Moody algebras of rank 2

ABSTRACTLet 𝔤 be a hyperbolic Kac–Moody algebra of rank 2, and set , where are the fundamental weights for 𝔤; note that λ is neither dominant nor antidominant. Let 𝔹(λ) be the crystal of all Lakshmibai–Seshadri paths of shape λ. We prove that (the crystal graph of) 𝔹(λ) is connected. Furthermore, we give an explicit description of Lakshmibai–Seshadri paths of shape λ.

Lakshmibai-Seshadri Paths and Non-Symmetric Cauchy Identity

We give a simple crystal theoretic interpretation of the Lascoux’s expansion of a non-symmetric Cauchy kernel ${\prod }_{i+ j \leq n + 1}(1-x_{i}y_{j})^{-1}$ , which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.

Specialization of nonsymmetric Macdonald polynomials at 𝑡=∞ and Demazure submodules of level-zero extremal weight modules

In this paper, we give a representation-theoretic interpretation of the specialization E w ∘ λ ( q , ∞ ) E_{w_\circ \lambda } (q,\infty ) of the nonsymmetric Macdonald polynomial E w ∘ λ ( q , t ) E_{w_\circ \lambda }(q,t) at t = ∞ t=\infty in terms of the Demazure submodule V w ∘ − ( λ ) V_{w_\circ }^- (\lambda ) of the level-zero extremal weight module V ( λ ) V(\lambda ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ \lambda is a dominant integral weight, and w ∘ w_\circ denotes the longest element in the finite Weyl group W W . Also, for each x ∈ W x \in W , we obtain a combinatorial formula for the specialization E x λ ( q , ∞ ) E_{x \lambda } (q, \infty ) at t = ∞ t=\infty of the nonsymmetric Macdonald polynomial E x λ ( q , t ) E_{x \lambda } (q,t) and also a combinatorical formula for the graded character gch ⁡ V x − ( λ ) \operatorname {gch} V_{x}^- (\lambda ) of the Demazure submodule V x − ( λ ) V_{x}^- (\lambda ) of V ( λ ) V(\lambda ) . Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ \lambda .

A UNIFORM MODEL FOR KIRILLOV–RESHETIKHIN CRYSTALS III: NONSYMMETRICMACDONALD POLYNOMIALS AT t = 0 AND DEMAZURE CHARACTERS

We establish the equality of the specialization $E_{w\lambda}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{w\lambda}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(\lambda)$ of a certain Demazure-type submodule $U_{w}^{+}(\lambda)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $\lambda$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_{\lambda}(x;q,0)$ of the symmetric Macdonald polynomial $P_{\lambda}(x;q,t)$ at $t=0$ and the graded character of a tensor product of single-column Kirillov--Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of a nonsymmetric Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model.

Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

In this paper, we give a characterization of the crystal bases $\mathcal{B}_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of Demazure submodules $V_{x}^{+}(\lambda)$, $x \in W_{\mathrm{af}}$, of a level-zero extremal weight module $V(\lambda)$ over a quantum affine algebra $U_{q}$, where $\lambda$ is an arbitrary level-zero dominant integral weight, and $W_{\mathrm{af}}$ denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis $\mathcal{B}(\lambda)$ of the level-zero extremal weight module $V(\lambda)$ and the crystal $\mathbb{B}^{\frac{\infty}{2}}(\lambda)$ of semi-infinite Lakshmibai-Seshadri paths of shape $\lambda$, which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule $V_{w_0}^{+}(\lambda)$ in terms of the specialization at $t=0$ of the symmetric Macdonald polynomial $P_{\lambda}(x\,;\,q,\,t)$.

Semi-infinite Lakshmibai–Seshadri path model for level-zero extremal weight modules over quantum affine algebras

We introduce semi-infinite Lakshmibai–Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of the previous result due to Naito and Sagaki (which uses Littelmann's Lakshmibai–Seshadri paths), in which the level-zero dominant integral weight is assumed to be a positive-integer multiple of a level-zero fundamental weight.

Lakshmibai–Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations

AbstractWe give an interpretation of the energy function and classically restricted one-dimensional sums associated to tensor products of level-zero fundamental representations of quantum affine algebras in terms of Lakshmibai–Seshadri paths of level-zero shape.

Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero shape

Crystal structure on the set of Lakshmibai-Seshadri paths of an arbitrary level-zero shape

Mullineux involution and twisted affine Lie algebras

We use Naito and Sagaki's work [S. Naito, D. Sagaki, Lakshmibai–Seshadri paths fixed by a diagram automorphism, J. Algebra 245 (2001) 395–412; S. Naito, D. Sagaki, Standard paths and standard monomials fixed by a diagram automorphism, J. Algebra 251 (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to study the partitions fixed by Mullineux involution. We characterize the set of Mullineux-fixed partitions in terms of crystal graphs of basic representations of twisted affine Lie algebras of type A 2 ℓ ( 2 ) and of type D ℓ + 1 ( 2 ) . We set up bijections between the set of symmetric partitions and the set of partitions into distinct parts. We propose a notion of double restricted strict partitions. Bijections between the set of restricted strict partitions (respectively, the set of double restricted strict partitions) and the set of Mullineux-fixed partitions in the odd case (respectively, in the even case) are obtained.

Crystal of Lakshmibai-Seshadri paths associated to an integral weight of level zero for an affine Lie algebra

Let λ=∑i∈I0miϖi with mi ∈ ℤ≥0 be an integral weight of level zero, where the ϖi⁠, i ∈ I0, are the level-zero fundamental weights for an affine Lie algebra g over ℚ, and let B(λ) be the crystal (with weight lattice P) of all Lakshmibai-Seshadri paths of shape λ. We define a crystal B(λ)cl (with weight lattice Pcl ≔ P/(ℚ δ∩ P)) to be the set of paths cl(π) : [0, 1] → ℚ ⊗ℤPcl for π ∈ B(λ), where the path cl(π) is defined to be the composition of the path π :[0,1] → ℚ ⊗ℤP with the canonical projection cl:ℚ⊗ℤP↠ℚ⊗ℤPcl⁠. We prove that this crystal B(\λ)cl is isomorphic to the tensor product ⊗i∈I0(B(ϖi)cl)⊗mi of the crystals B(ϖi)cl⁠, i ∈ I0, as a crystal with weight lattice Pcl. As a corollary, we obtain that for each integral weight λ ∈ P of level zero, the crystal B(λ)cl is isomorphic, as a crystal with weight lattice Pcl, to the crystal base of a (certain) tensor product of the fundamental modules W(ϖi)⁠, i ∈ I0, of level zero over the quantized universal enveloping algebra Uq′(g) of g over ℚ(q) with weight lattice Pcl.

Path model for a level-zero extremal weight module over a quantum affine algebra II

Let ϖ i be a level-zero fundamental weight for an affine Lie algebra g over Q , and let B ( ϖ i ) be the crystal of all Lakshmibai–Seshadri paths of shape ϖ i . First, we prove that the crystal graph of B ( ϖ i ) is connected. By combining this fact with the main result of our previous work, we see that B ( ϖ i ) is, as a crystal, isomorphic to the crystal base B ( ϖ i ) of the extremal weight module V ( ϖ i ) over a quantum affine algebra U q ( g ) over Q ( q) of extremal weight ϖ i . Next, we obtain an explicit description of the decomposition of the crystal B ( m ϖ i ) of all Lakshmibai–Seshadri paths of shape m ϖ i into connected components. Furthermore, we prove that B ( m ϖ i ) is, as a crystal, isomorphic to the crystal base B ( m ϖ i ) of the extremal weight module V ( m ϖ i ) over U q ( g ) of extremal weight m ϖ i .

Crystal Bases, Path Models, and a Twining Character Formula for Demazure Modules

We give a combinatorial proof of a twining character formula for Demazure modules, by combining the isomorphism theorem between path models and crystal bases with our previous result about Lakshmibai-Seshadri paths fixed by a diagram automorphism.