arXiv subjects

Leonardo C. Mihalcea

Publications and source records attributed to Leonardo C. Mihalcea.

At least 19 recordsLinked to original sources

arXiv2026

Positivity of minuscule quantum K-theory

We prove that the Schubert structure constants of the quantum $K$-theory ring of any minuscule flag variety or quadric hypersurface have signs that alternate with codimension. We also prove that the powers of the deformation parameter $q$ that occur in the product of two Schubert classes in the quantum cohomology or quantum $K$-theory ring of a cominuscule flag variety form an integer interval. Our proofs are based on several new results, including an explicit description of the most general non-empty intersection of two Schubert varieties in an arbitrary flag manifold, and a computation of the cohomology groups of any negative line bundle restricted to a Richardson variety in a cominuscule flag variety. We also give a type-uniform proof of the quantum-to-classical theorem, which asserts that the (3-point, genus 0) Gromov-Witten invariants of any cominuscule flag variety are classical triple-intersection numbers on an associated flag variety. Finally, we prove several new results about the geometry and combinatorics related to this theorem.

math.AG↗

arXiv2025

A Nakayama result for the quantum K theory of homogeneous spaces

We prove that the ideal of relations in the (equivariant) quantum K ring of a homogeneous space is generated by quantizations of each of the generators of the ideal in the classical (equivariant) K ring. This extends to quantum K theory a result of Siebert and Tian in quantum cohomology. We illustrate this technique in the case of the quantum K ring of partial flag manifolds, using a set of quantum K Whitney relations conjectured by the authors, and recently proved by Huq-Kuruvilla.

math.AG↗

arXiv2025

Toda-Type Presentations for the Quantum K Theory of Partial Flag Varieties

We prove a determinantal, Toda-type, presentation for the equivariant K theory of a partial flag variety ${\rm Fl}(r_1, \dots, r_k;n)$. The proof relies on pushing forward the Toda presentation obtained by Maeno, Naito and Sagaki for the complete flag variety ${\rm Fl}(n)$, via Kato's ${\rm K}_T({\rm pt})$-algebra homomorphism from the quantum K ring of ${\rm Fl}(n)$ to that of ${\rm Fl}(r_1, \dots, r_k;n)$. Starting instead from the Whitney presentation for ${\rm Fl}(n)$, we show that the same pushforward technique gives a recursive formula for polynomial representatives of quantum K Schubert classes in any partial flag variety which do not depend on quantum parameters. In an appendix, we include another proof of the Toda presentation for the equivariant quantum K ring of ${\rm Fl}(n)$, following Anderson, Chen, and Tseng, which is based on the fact that the ${\rm K}$-theoretic $J$-function is an eigenfunction of the finite difference Toda Hamiltonians.

math.AG↗

arXiv2025

Quantum $K$-theoretic divisor axiom for flag manifolds

We prove an identity for (torus-equivariant) 3-point, genus 0, $K$-theoretic Gromov-Witten invariants of flag manifolds $G/P$, which can be thought of as a replacement for the ``divisor axiom'' in their (torus-equivariant) quantum $K$-theory. This identity enables us to compute these invariants when two insertions are Schubert classes and the other a Schubert divisor class. Our type-independent proof utilizes the Chevalley formula for the (torus-equivariant) quantum $K$-theory ring of flag manifolds, which computes multiplications by Schubert divisor classes in terms of the quantum Bruhat graph.

math.QA↗

arXiv2025

Quantum K--theory of Grassmannians from a Yang-Baxter algebra

In an earlier paper, two of the authors defined a $5$-vertex Yang-Baxter algebra (a Hopf algebra) which acts on the sum of the equivariant quantum K-rings of Grassmannians $\mathrm{Gr}(k;n)$, where $k$ varies from $0$ to $n$. We construct geometrically defined operators on quantum K-rings describing this action. In particular, the $R$-matrix defining the Yang-Baxter algebra corresponds to the left Weyl group action. Most importantly, we use the `quantum=classical' statement for the quantum K-theory of Grassmannians to prove an explicit geometric interpretation of the action of generators of the Yang-Baxter algebra. The diagonal entries of the monodromy matrix are given by quantum K-multiplications by explicitly defined classes, and the off-diagonal entries by certain push-pull convolutions. We use this to find a quantization of the classes of fixed points in the quantum K-rings, corresponding to the Bethe vectors of the Yang-Baxter algebra. On each of the quantum K-rings, we prove that the two Frobenius structures (one from geometry, and the other from the integrable system construction) coincide. We discuss several applications, including an action of the extended affine Weyl group on the quantum K-theory ring (extending the Seidel action), a quantum version of the localization map (which is a ring homomorphism with respect to the quantum K-product), and a graphical calculus to multiply by Hirzebruch $λ_y$ classes of the dual of the tautological quotient bundle. In an Appendix we illustrate our results in the case when $n=2$.

math.AG↗

arXiv2025

Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes

We prove a Chevalley formula to multiply the motivic Chern classes of Schubert cells in a generalized flag manifold $G/P$ by the class of any line bundle $\mathcal{L}_λ$. Our formula is given in terms of the $λ$-chains of Lenart and Postnikov. Its proof relies on a change of basis formula in the affine Hecke algebra due to Ram, and on the Hecke algebra action on torus-equivariant K-theory of the complete flag manifold $G/B$ via left Demazure--Lusztig operators. We revisit some wall-crossing formulae for the stable envelopes in $T^*(G/B)$. We use our Chevalley formula, and the equivalence between motivic Chern classes of Schubert cells and K-theoretic stable envelopes in $T^*(G/B)$, to give formulae for the change of polarization, and for the change of slope for stable envelopes. We prove several additional applications, including Serre, star, and Dynkin, dualities of the Chevalley coefficients, new formulae for the Whittaker functions, and for the Hall--Littlewood polynomials. We also discuss positivity properties of Chevalley coefficients, and properties of the coefficients arising from multiplication by minuscule weights.

math.AG↗

arXiv2024

Quantum K Whitney relations for partial flag varieties

In a recent paper, we stated conjectural presentations for the equivariant quantum K ring of partial flag varieties, motivated by physics considerations. In this companion paper, we analyze these presentations mathematically. We start by proving a Nakayama type result for quantum K theory: if the conjectured set of relations deforms a complete set of relations of the classical K theory ring, then they must form a complete set of relations for the quantum K ring. We prove the conjectured presentation in the case of the incidence varieties, and we show that if a quantum K divisor axiom holds (as conjectured by Buch and Mihalcea), then the conjectured presentation also holds for the complete flag variety. Finally, we briefly revisit the change of variables relating the mathematics and physics presentations.

math.AG↗

arXiv2024

Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles

We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch $λ_y$ classes of the tautological bundles. In physics, the $λ_y$ classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. Along the way, we calculate K-theoretic Gromov-Witten invariants of wedge powers of the tautological bundles on $\mathrm{Gr}(k;n)$, using the `quantum=classical' statement.

math.AG↗

arXiv2024

Hook formula for Coxeter groups via the twisted group ring

We use Kostant and Kumar's twisted group ring and its dual to formulate and prove a generalization of Nakada's colored hook formula for any Coxeter groups. For dominant minuscule elements of the Weyl group of a Kac--Moody algebra, this provides another short proof of Nakada's colored hook formula.

math.RT↗

arXiv2023

Characteristic cycles of IH sheaves of simply laced minuscule Schubert varieties are irreducible

Let $G/P$ be a complex cominuscule flag manifold of type $E_6,E_7$. We prove that each characteristic cycle of the intersection homology (IH) complex of a Schubert variety in $G/P$ is irreducible. The proof utilizes an earlier algorithm by the same authors which calculates local Euler obstructions, then proceeds by direct computer calculation using Sage. This completes to the exceptional Lie types the characterization of irreducibility of IH sheaves of Schubert varieties in cominuscule $G/P$ obtained by Boe and Fu. As a by-product, we also obtain that the Mather classes, and the Chern-Schwartz-MacPherson classes of Schubert cells in cominuscule $G/P$ of type $E_6,E_7$, are strongly positive.

math.AG↗

arXiv2022

From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes

The equivariant motivic Chern class of a Schubert cell in a `complete' flag manifold $X=G/B$ is an element in the equivariant K theory ring of $X$ to which one adjoins a formal parameter $y$. In this paper we prove several `folklore results' about the motivic Chern classes, including finding specializations at $y=-1$ and $y=0$; the coefficient of the top power of $y$; how to obtain Chern-Schwartz-MacPherson (CSM) classes as leading terms of motivic classes; divisibility properties of the Schubert expansion of motivic Chern classes. We collect several conjectures about the positivity, unimodality, and log concavity of CSM and motivic Chern classes of Schubert cells, including a conjectural positivity of structure constants of the multiplication of Poincaré duals of CSM classes. In addition, we prove a `star duality' for the motivic Chern classes. We utilize the motivic Chern transformation to define two equivariant variants of the Hirzebruch transformation, which appear naturally in the Grothendieck-Hirzebruch-Riemann-Roch formalism. We utilize the Demazure-Lusztig recursions from the motivic Chern class theory to find similar recursions giving the Hirzebruch classes of Schubert cells, their Poincar{é} duals, and their Segre versions. We explain the functoriality properties needed to extend the results to `partial' flag manifolds $G/P$.

math.AG↗

arXiv2022

Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells

Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex projective manifold $X$, the homogenized, torus equivariant CSM class of a constructible function $φ$ is the restriction of the characteristic cycle of $φ$ via the zero section of the cotangent bundle of $X$. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize $X$ to be a (generalized) flag manifold $G/B$. In this case CSM classes are determined by a Demazure-Lusztig (DL) operator. We prove a `Hecke orthogonality' of CSM classes, determined by the DL operator and its Poincar{é} adjoint. We further use the theory of holonomic $\mathcal{D}_X$-modules to show that the characteristic cycle of a Verma module, restricted to the zero section, gives the CSM class of the corresponding Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov's stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rim{á}nyi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the expansion in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds $G/P$.

math.AG↗

arXiv2022

Whittaker functions from motivic Chern classes

We prove a `motivic' analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized flag manifold $G/B$ multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure-Lusztig (D-L) operators, recovers formulas found by Brubaker, Bump and Licata for the Iwahori-Whittaker functions of the principal series representation of a $p$-adic group. In particular, we obtain a new proof of the classical Casselman-Shalika formula for the spherical Whittaker function. The proofs are based on localization in equivariant K theory, and require a geometric interpretation of how the Hecke dual (or inverse) of a D-L operator acts on the class of a point. We prove that the Hecke dual operators give Grothendieck-Serre dual classes of the motivic classes, a result which might be of independent interest. In an Appendix joint with Dave Anderson we show that if the line bundle is trivial, we recover a generalization of a classical formula by Kostant, Macdonald, Shapiro and Steinberg for the Poincar{é} polynomial of $G/B$; the generalization we consider is due to Akyıldız and Carrell and replaces $G/B$ by any smooth Schubert variety.

math.AG↗

arXiv2022

Hook formulae from Segre-MacPherson classes

Nakada's colored hook formula is a vast generalization of many important formulae in combinatorics, such as the classical hook length formula and the Peterson's formula for the number of reduced expressions of minuscule Weyl group elements. In this paper, we utilize cohomological properties of Segre-MacPherson classes of Schubert cells and varieties to prove a generalization of a cohomological version of Nakada's formula, in terms of smoothness properties of Schubert varieties. A key ingredient in the proof is the study of a decorated version of the Bruhat graph. Summing over weighted paths of this graph give the terms in the generalized Nakada's formula, and also provide algorithms to calculate structure constants of multiplications of Segre-MacPherson classes of Schubert cells. For simply laced Weyl groups, we also show the equality of `skew' and `straight' Nakada's formulae. This utilizes a criterion for smoothness in terms of excited diagrams of heaps of minuscule elements, which might be of independent interest.

math.CO↗

arXiv2022

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

Motivic Chern classes are elements in the K-theory of an algebraic variety $X$, depending on an extra parameter $y$. They are determined by functoriality and a normalization property for smooth $X$. In this paper we calculate the motivic Chern classes of Schubert cells in the (equivariant) K-theory of flag manifolds $G/B$. We show that the motivic class of a Schubert cell is determined recursively by the Demazure-Lusztig operators in the Hecke algebra of the Weyl group of $G$, starting from the class of a point. The resulting classes are conjectured to satisfy a positivity property. We use the recursions to give a new proof that they are equivalent to certain K-theoretic stable envelopes recently defined by Okounkov and collaborators, thus recovering results of Fehér, Rimányi and Weber. The Hecke algebra action on the K-theory of the Langlands dual flag manifold matches the Hecke action on the Iwahori invariants of the principal series representation associated to an unramified character for a group over a nonarchimedean local field. This gives a correspondence identifying the duals of the motivic Chern classes to the standard basis in the Iwahori invariants, and the fixed point basis to Casselman's basis. We apply this correspondence to prove two conjectures of Bump, Nakasuji and Naruse concerning factorizations and holomorphy properties of the coefficients in the transition matrix between the standard and the Casselman's basis.

math.AG↗

arXiv2021

Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory

We study the Demazure-Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K theory), in any partial flag manifold. Along the way we advertise many properties of the left and right divided difference operators in cohomology and K theory, and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K theory, generating Schubert classes, and satisfying a Leibniz rule compatible with the quantum product.

math.AG↗

arXiv2020

Mather classes and conormal spaces of Schubert varieties in cominuscule spaces

Let $G/P$ be a complex cominuscule flag manifold. We prove a type independent formula for the torus equivariant Mather class of a Schubert variety in $G/P$, and for a Schubert variety pulled back via the natural projection $G/Q \to G/P$. We apply this to find formulae for the local Euler obstructions of Schubert varieties, and for the torus equivariant localizations of the conormal spaces of these Schubert varieties. We conjecture positivity properties for the local Euler obstructions and for the Schubert expansion of Mather classes. We check the conjectures in many cases, by utilizing results of Boe and Fu about the characteristic cycles of the intersection homology sheaves of Schubert varieties. We also conjecture that certain `Mather polynomials' are unimodal in general Lie type, and log concave in type A.

math.AG↗

arXiv2020

Positivity of Segre-MacPherson classes

Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on $X$ with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.

math.AG↗