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Daniel Orr

Publications and source records attributed to Daniel Orr.

At least 19 recordsLinked to original sources
arXiv2025

Combinatorial proof of a permuted basement Macdonald polynomial identity

A well-known and fundamental property of the Macdonald polynomials $P_\lambda(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, we refine the Concha-Lapointe identity to a sub-family of Alexandersson's permuted basement Macdonald polynomials and give a combinatorial proof of the refined identity. We show also that the Concha-Lapointe identity is equivalent to the assertion that (normalized) partially symmetric Macdonald polynomials are fixed under the Kazhdan-Lusztig involution.

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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.

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arXiv2024

Stable-limit partially symmetric Macdonald functions and parabolic flag Hilbert schemes

The modified Macdonald functions $\widetilde{H}_{\mu}$ are fundamental objects in modern algebraic combinatorics. Haiman showed that there is a correspondence between the $(\mathbb{C}^{*})^2$-fixed points $I_{\mu}$ of the Hilbert schemes $\mathrm{Hilb}_{n}(\mathbb{C}^2)$ and the functions $\widetilde{H}_{\mu}$ realizing a derived equivalence between $(\mathbb{C}^{*})^2$-equivariant coherent sheaves on $\mathrm{Hilb}_{n}(\mathbb{C}^2)$ and $(\mathfrak{S}_n \times (\mathbb{C}^{*})^2)$-equivariant coherent sheaves on $(\mathbb{C}^2)^n.$ Carlsson--Gorsky--Mellit introduced a larger family of smooth projective varieties $\mathrm{PFH}_{n,n-k}$ called the parabolic flag Hilbert schemes. They showed that an algebra $\mathbb{B}_{q,t}$, directly related to the double Dyck path algebra $\mathbb{A}_{q,t}$ employed in Carlsson--Mellit's proof of the Shuffle Theorem, acts naturally on the $(\mathbb{C}^{*})^2$-equivariant K-theory $U_{\bullet}$ of these spaces and, moreover, there is a $\mathbb{B}_{q,t}$-isomorphism $\Phi: U_{\bullet} \rightarrow V_{\bullet}$ where $V_{\bullet}$ is the polynomial representation. The isomorphism $\Phi: U_{\bullet} \rightarrow V_{\bullet}$ is known to extend Haiman's correspondence. In this paper, we explicitly compute the images $\Phi(H_{\mu,w})$ of the normalized $(\mathbb{C}^{*})^2$-fixed point classes $H_{\mu,w}$ of the spaces $\mathrm{PFH}_{n,n-k}$ and show they agree with the modified partially symmetric Macdonald polynomials $\widetilde{H}_{(\lambda|\gamma)}$ introduced by Goodberry-Orr, confirming their prior conjecture. We use this result to give an explicit formula for the action of the involution $\mathcal{N}$ on $V_{\bullet}.$

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arXiv2023

A Geometric Realization of Partially-Symmetric Macdonald Polynomials

We formulate a precise conjecture relating integral form partially-symmetric Macdonald polynomials and the parabolic flag Hilbert schemes of Carlsson, Gorsky, and Mellit. This extends, in an explicit fashion, Haiman's realization of modified Macdonald symmetric functions via Hilbert schemes of points in the plane. As evidence for our conjecture we prove that it is compatible with the action of certain elements in Carlsson and Mellit's algebra $\mathbb{A}_{t,q}$, including degree $1$ Pieri formulas.

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arXiv2023

Wreath Macdonald polynomials, a survey

Wreath Macdonald polynomials arise from the geometry of $\Gamma$-fixed loci of Hilbert schemes of points in the plane, where $\Gamma$ is a finite cyclic group of order $r\ge 1$. For $r=1$, they recover the classical (modified) Macdonald symmetric functions through Haiman's geometric realization of these functions. The existence, integrality, and positivity of wreath Macdonald polynomials for $r>1$ was conjectured by Haiman and first proved in work of Bezrukavnikov and Finkelberg by means of an equivalence of derived categories. Despite the power of this approach, a lack of explicit tools providing direct access to wreath Macdonald polynomials -- in the spirit of Macdonald's original works -- has limited progress in the subject. A recent result of Wen provides a remarkable set of such tools, packaged in the representation theory of quantum toroidal algebras. In this article, we survey Wen's result along with the basic theory of wreath Macdonald polynomials, including its geometric foundations and the role of bigraded reflection functors in the construction of wreath analogs of the $\nabla$ operator. We also formulate new conjectures on the values of important constants arising in the theory of wreath Macdonald $P$-polynomials. A variety of examples are used to illustrate these objects and constructions throughout the paper.

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arXiv2023

Peter-Weyl theorem for Iwahori groups and highest weight categories

We study the algebra of functions on the Iwahori group via the category of graded bounded representations of its Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized Weyl modules. Using this identification we express the characters of the standard and costandard objects in terms of specialized nonsymmetric Macdonald polynomials. We also prove that our category of interest admits a generalized highest weight structure (known as stratified structure). We show, more generally, that such a structure on a category of representations of a Lie algebra implies the Peter-Weyl type theorem for the corresponding algebraic group. In the Iwahori case, standard filtrations of indecomposable projective objects correspond to new ``reciprocal'' Macdonald-type identities.

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arXiv2023

Nonsymmetric $q$-Cauchy identity and representations of the Iwahori algebra

The $t=0$ specialization of the Mimachi-Noumi Cauchy-type identity rewrites certain infinite product in terms of specialized nonsymmetric Macdonald polynomials of type $GL_n$. We interpret the infinite product as a character of the space of functions on a certain matrix space. We show that the space of functions admits a filtration such that the graded pieces are isomorphic to the tensor products of certain generalized global Weyl modules of the Iwahori algebra. We identify the characters of the graded pieces with the terms of the specialized Mimachi-Noumi formula. We conjecture the existence of an analogous filtration on the space of functions on the Iwahori group for all simple Lie algebras and prove the conjecture for $SL_n$. Our construction can be seen as a current algebra extension of the van der Kallen filtration on functions on a Borel subgroup.

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arXiv2022

Wreath Macdonald operators

We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald $P$-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our operators arise from integral formulas for the action of the horizontal Heisenberg subalgebra in the vertex representation of the corresponding quantum toroidal algebra

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arXiv2022

Identities of inverse Chevalley type for graded characters of level-zero Demazure submodules over quantum affine algebras of type C

We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type $C$. These identities express the product $e^{\mu} \, \mathrm{gch} \, V_{x}^{-}(\lambda)$ of the (one-dimensional) character $e^{\mu}$, where $\mu$ is a (not necessarily dominant) minuscule weight, with the graded character $\mathrm{gch} \, V_{x}^{-}(\lambda)$ of the level-zero Demazure submodule $V_{x}^{-}(\lambda)$ over the quantum affine algebra $U_{\mathsf{q}}(\mathfrak{g}_{\mathrm{af}})$ as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas in the torus-equivariant $K$-group of the semi-infinite flag manifold $\mathbf{Q}_{G}$ associated to a connected, simply-connected and simple algebraic group $G$ of type $C$. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that $\mu$ is a standard basis element $\varepsilon_{k}$ in the weight lattice $P$ of $G$.

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arXiv2020

Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type and equivariant scalars $e^{\lambda}$, where $\lambda$ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply-laced type, except for type $E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. As such, our formula also provides an explicit determination of all nonsymmetric $q$-Toda operators for minuscule weights in ADE type.

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arXiv2020

Equivariant $K$-theory of the semi-infinite flag manifold as a nil-DAHA module

The equivariant $K$-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a $q$-Heisenberg algebra. The action of the latter generates a free submodule of rank $|W|$, where $W$ is the (finite) Weyl group. We show that this submodule is stable under the nil-DAHA, which enables one to express the nil-DAHA action in terms of $W\times W$ matrices over the $q$-Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric $q$-Toda system introduced by Cherednik and the author.

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arXiv2018

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

We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) quantum $K$-group of finite-dimensional flag manifolds. Our proof of these formulas is based on standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, which is established in our previous work, and also uses a string property of Demazure-like subsets of the crystal basis of a level-zero extremal weight module over a quantum affine algebra.

math.QA↗
arXiv2017

Walk algebras, distinguished subexpressions, and point counting in Kac-Moody flag varieties

We study walk algebras and Hecke algebras for Kac-Moody root systems. Each choice of orientation for the set of real roots gives rise to a corresponding "oriented" basis for each of these algebras. We show that the notion of distinguished subexpression naturally arises when studying the transition matrix between oriented bases. We then relate these notions to the geometry of Kac-Moody flag varieties and Bott-Samelson varieties. In particular, we show that the number of points over a finite field in certain intersections of these varieties is given by change of basis coefficients between oriented bases of the Hecke algebra. Using these results we give streamlined derivations of Deodhar's formula for $R$-polynomials and point-counting formulas for specializations of nonsymmetric Macdonald polynomials $E_\lambda(\mathsf{q},t)$ at $\mathsf{q}=0,\infty$.

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arXiv2017

Quiver Hall-Littlewood functions and Kostka-Shoji polynomials

For any triple $(i,a,\mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $\mu$, we define a quiver current $H^{(i,a)}_\mu$ acting on the tensor power $\Lambda^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},\mu(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ as the result of acting on $1\in\Lambda^Q$ by the corresponding sequence of quiver currents. The quiver Kostka-Shoji polynomials are the expansion coefficients of $H^{\mathbf{i},\mathbf{a}}_{\mu(\bullet)}$ in the tensor Schur basis. These polynomials include the Kostka-Foulkes polynomials and parabolic Kostka polynomials (Jordan quiver) and the Kostka-Shoji polynomials (cyclic quiver) as special cases. We show that the quiver Kostka-Shoji polynomials are graded multiplicities in the equivariant Euler characteristic of a vector bundle on Lusztig's convolution diagram determined by the sequences $\mathbf{i},\mathbf{a}$. For certain compositions of currents we conjecture higher cohomology vanishing of the associated vector bundle on Lusztig's convolution diagram. For quivers with no branching we propose an explicit positive formula for the quiver Kostka-Shoji polynomials in terms of catabolizable multitableaux. We also relate our constructions to $K$-theoretic Hall algebras, by realizing the quiver Kostka-Shoji polynomials as natural structure constants and showing that the quiver currents provide a symmetric function lifting of the corresponding shuffle product. In the case of a cyclic quiver, we explain how the quiver currents arise in Saito's vertex representation of the quantum toroidal algebra of type $\mathfrak{sl}_r$.

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arXiv2016

Stochastic higher spin six vertex model and q-TASEPs

We present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the Macdonald processes hierarchy. First, we show how Macdonald $q$-difference operators with $t=0$ (an algebraic tool crucial for studying the corresponding Macdonald processes) can be utilized to get $q$-moments of the height function $\mathfrak{h}$ in the higher spin six vertex model first computed in arXiv:1601.05770 using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the value of $\mathfrak{h}$ at an arbitrary point $(N+1,T)\in\mathbb{Z}_{\ge2}\times\mathbb{Z}_{\ge1}$ has the same distribution as the last component $\lambda_N$ of a random partition under a specific $t=0$ Macdonald measure. On the other hand, it is known that $\mathbf{x}_N:=\lambda_N-N$ can be identified with the location of the $N$th particle in a certain discrete time $q$-TASEP started from the step initial configuration. The second construction we present is a coupling of this $q$-TASEP and the higher spin six vertex model (with the step Bernoulli boundary condition) along time-like paths providing an independent probabilistic explanation of the equality of $\mathfrak{h}(N+1,T)$ and $\mathbf{x}_N+N$ in distribution. Combined with the identification of averages of observables between the stochastic higher spin six vertex model and Schur measures (which are $t=q$ Macdonald measures) obtained recently in arXiv:1608.01553, this produces GUE Tracy--Widom asymptotics for a discrete time $q$-TASEP with the step initial configuration and special jump parameters.

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