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Syu Kato

Publications and source records attributed to Syu Kato.

At least 19 recordsLinked to original sources
arXiv2025

Categorification of $k$-Schur functions and refined Macdonald positivity

We characterize the $k$-Schur functions as the graded characters of simple objects in an additive module category. This confirms a set of conjectures formulated in the Ph.D. thesis of Chen, written under the direction of Mark Haiman, and thereby establishes the algebraic framework proposed therein. As a consequence, we deduce that the modified Macdonald polynomials are $k$-Schur positive, thus realizing the original motivation behind the definition of the $k$-Schur functions by Lapointe, Lascoux, and Morse. Our approach builds on our previous work on the algebraic and geometric realization of Catalan symmetric functions, which encompasses both the $k$-Schur functions and the Hall--Littlewood functions.

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arXiv2024

A geometric realization of the chromatic symmetric function of a unit interval graph

Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. ${\bf 295}$ (2016), ${\bf 329}$ (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety $\mathscr X_\Psi$ defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on this, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Shareshian-Wachs conjecture.

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arXiv2024

Kostka polynomials of $G(\ell,1,m)$

For each integers $\ell > 1$ and $n \ge m \ge 1$, we prove an equivalence between the category of polynomial modules over a paraholic subalgebra $\mathfrak p$ of an affine Lie algebra of $\mathfrak{gl}(n\ell)$ and the module category of the smash product algebra $A$ of the complex reflection group $G(\ell,1,m)$ with $\mathbb C [X_1,\ldots,X_m]$. Then, we transfer the collection of $\mathfrak p$-modules considered in [Feigin-Makedonskyi-Khoroshkhin, arXiv:2311.12673] to $A$. Applying the Lusztig-Shoji algorithm [Shoji, Invent. Math. ${\bf 74}$ (1983)] (or rather its homological variant [K. Ann. Sci. ENS ${\bf 48}$(5) (2015)]), we conclude that the multiplicity counts of these modules yield the Kostka polynomials attached to the limit symbols in the sense of [Shoji, ASPM ${\bf 40}$ (2004)]. This particularly settles a conjecture of Shoji [${\it loc. cit.}$ \S 3.13] and answers a question in [Shoji, Sci. China Math. ${\bf 61}$ (2018)].

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arXiv2023

A geometric realization of Catalan functions

We construct smooth projective varieties $\sX_\Psi$, each of which compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\GL(n)$. Each variety $\sX_\Psi$ carries a natural family of line bundles whose spaces of global sections realize, as graded characters, the Catalan functions introduced in Chen's thesis and further studied by Blasiak, Morse, Pun, and Summers. Using the geometry of $\sX_\Psi$, we prove the Chen--Haiman vanishing conjecture and confirm the tame case of the Blasiak--Morse--Pun vanishing conjecture. We further establish the Shimozono--Weyman monotonicity conjectures.

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arXiv2022

Higher level BGG reciprocity for current algebras

We establish a higher-level analogue of the Bernstein--Gelfand--Gelfand (BGG) reciprocity for twisted current algebras at every positive integer level, extending the level-one results of Bennett et al. We show that these reciprocity relations form a hierarchy linking adjacent levels, thereby providing a systematic framework to relate representations across arbitrary levels by iteration. Our formulation of the BGG reciprocity naturally includes infinite-dimensional (thick) Demazure modules. Consequently, our main theorem provides the first effective means to analyze the module-theoretic structure of thick Demazure modules at any level. As an application, we realize level-restricted generalized Kostka polynomials as branching polynomials in the language of symmetric polynomials. The resulting level-dependent picture also naturally incorporates theta functions and modular forms. Furthermore, we prove that every finite-dimensional Demazure module admits a filtration by Demazure modules of the underlying simple Lie algebra, resolving a long-standing speculation in non-simply-laced types.

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arXiv2021

Categorification of DAHA and Macdonald polynomials

We describe a categorification of the Double Affine Hecke Algebra (${\mathcal{H}\kern -.4em\mathcal{H}}$) associated with an affine Lie algebra $\widehat{\mathfrak{g}}$, including a categorification of the polynomial representation and Macdonald polynomials. Our categorification results are presented in the derived setting, focusing on the derived category of graded modules over the Lie superalgebra ${\mathfrak I}[\xi]$, where ${\mathfrak I} \subset \widehat{\mathfrak{g}}$ is the Iwahori subalgebra of the affine Lie algebra and $\xi$ is a formal odd variable. First, we show that the compositions of induction and restriction functors associated with minimal parabolic subalgebras ${\mathfrak{p}}_{i}$ categorify the Demazure operators $T_i + 1 \in {\mathcal{H}\kern -.4em\mathcal{H}}$, ensuring that all algebraic relations of $T_i$ have categorical interpretations. Second, for each dominant weight $\lambda$ we introduce a complex ${\mathbb{EM}}_{\lambda}$ of ${\mathfrak{I}}[\xi]$-modules and a complex ${\mathbb{PM}}_{\lambda}$ of ${\mathfrak{g}}[z,\xi]$-modules, whose Euler characteristics are equal to nonsymmetric $E_{\lambda}$ and symmetric $P_{\lambda}$ Macdonald polynomials respectively. We illustrate our theory with the example $\mathfrak{g}=\mathfrak{sl}_2$ where we construct the cyclic representations of Lie superalgebra ${\mathfrak{I}}[\xi]$ such that their supercharacters coincide with certain normalizations of nonsymmetric Macdonald polynomials.

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arXiv2020

Symmetric functions and Springer representations

The characters of the (total) Springer representations are identified with the Green functions by Kazhdan [Israel J. Math. {\bf 28} (1977)], and the latter are identified with Hall-Littlewood's $Q$-functions by Green [Trans. Amer. Math. Soc. (1955)]. In this paper, we present a purely algebraic proof that the (total) Springer representations of $\mathop{GL} ( n )$ are $\mathrm{Ext}$-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.

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arXiv2020

Darboux coordinates on the BFM spaces

Bezrukavnikov-Finkelberg-Mirkovi\'c [Compos. Math. {\bf 141} (2005)] identified the equivariant $K$-group of an affine Grassmannian, that we refer as (the coordinate ring of) a BFM space \'a l\`a Teleman [Proc. ICM Seoul (2014)], with a version of Toda lattice. We give a new system of generators and relations of a certain localization of this space, that can be seen as a version of its Darboux coordinate. This establishes a conjecture in Finkelberg-Tymbaliuk [Progress in Math. {\bf 300} (2019)] that relates the BFM space of a connected reductive algebraic group with those of Levi subgroups.

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arXiv2019

On quantum $K$-groups of partial flag manifolds

We show that the equivariant small quantum $K$-group of a partial flag manifold is a quotient of that of the full flag manifold in a way that respects the Schubert classes. This is a $K$-theoretic analogue of the parabolic version of Peterson's theorem [Lam-Shimozono, Acta Math. {\bf 204} (2010)] that exhibits a different behavior from the case of quantum cohomology. Our quotient maps send some of the Novikov variables to $1$, and the geometric meaning of this specialization is unclear in quantum $K$-theory.

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arXiv2018

Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb K$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind)scheme structure, its projective coordinate ring has a $\mathbb Z$-model, and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. {\bf 371} no.2 (2018)]) when $\mathsf{char} \, \mathbb K =0$ or $\gg 0$, and the higher cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$. Some particular cases of these results play crucial roles in our proof [K, arXiv:1805.01718] of a conjecture by Lam-Li-Mihalcea-Shimozono [J. Algebra {\bf 513} (2018)] that describes an isomorphism between affine and quantum $K$-groups of a flag manifold.

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arXiv2018

Loop structure on equivariant $K$-theory of semi-infinite flag manifolds

We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\bf 146} (2010)] coincides with the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. {\bf 169} (2020)]. Then, we construct an explicit isomorphism between the equivariant $K$-group of a semi-infinite flag manifold with a suitably localized equivariant quantum $K$-group of the corresponding flag manifold. These exhibit a new framework to understand the ring structure of equivariant quantum $K$-theory and the Peterson isomorphism.

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arXiv2018

Nonsymmetric Rogers-Ramanujan sums and thick Demazure modules

We consider expansions of products of theta-functions associated with arbitrary root systems in terms of nonsymmetric Macdonald polynomials at $t=\infty$ divided by their norms. The latter are identified with the graded characters of Demazure slices, some canonical quotients of thick (upper) level-one Demazure modules, directly related to recent theory of generalized (nonsymmetric) global Weyl modules. The symmetric Rogers-Ramanujan-type series considered by Cherednik-Feigin were expected to have some interpretation of this kind; the nonsymmetric setting appeared necessary to achieve this. As an application, the coefficients of the nonsymmetric Rogers-Ramanujan series provide formulas for the multiplicities of the expansions of tensor products of level-one Kac-Moody representations in terms of Demazure slices.

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arXiv2017

A Weyl Module Stratification of Integrable Representations (with an appendix by Ryosuke Kodera)

We construct a filtration on integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each Weyl module there is given by a corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type $\mathsf{ADE}$ in terms of Schubert manifolds of thick affine Grassmanian, as predicted by Boris Feigin.

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arXiv2017

Frobenius splitting of thick flag manifolds of Kac-Moody algebras

We explain that the Pl\"ucker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag manifolds to the thick case. As a consequence, we provide a description of the global sections of line bundles of a thick Schubert variety as conjectured in Kashiwara-Shimozono [Duke Math. J. 148 (2009)]. This also yields the existence of a compatible basis of thick Demazure modules, and the projective normality of the thick Schubert varieties.

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arXiv2017

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

We study the nonsymmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the nonsymmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules.

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arXiv2017

Equivariant $K$-theory of semi-infinite flag manifolds and 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.

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arXiv2016

Demazure character formula for semi-infinite flag varieties

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of global sections of a semi-infinite Schubert variety. Moreover, we give geometric realizations of Feigin-Makedonskyi's generalized Weyl modules, and the $t = \infty$ specialization of non-symmetric Macdonald polynomials.

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