arXiv subjects

Arun Ram

Publications and source records attributed to Arun Ram.

At least 19 recordsLinked to original sources

arXiv2026

Formulas for Koornwinder polynomials

This paper provides formulas for Koornwinder polynomials in analogy with the creation formula, the alcove walk formula and the non-attacking fillings formula for the type $GL_n$ Macdonald polynomials. We state the creation formula in terms of the divided-difference operators used in Schubert calculus, and we use a box-greedy reduced word to reformulate the alcove walk formula in terms of uncompressed set-valued tableaux. Then two types of compression, ``around-the-end compression'' and ``across-the-$0$-gap compression'', are used to derive a formula for Koornwinder polynomials in terms of compressed set-valued tableaux. Throughout we work in the full generality of relative Koornwinder polynomials, which are the analogues of the permuted basement Macdonald polynomials used in the type $GL_n$ case.

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arXiv2026

A memorial tribute: Adriano Garsia (1928--2024)

Adriano Mario Garsia was born in Tunis on August 20, 1928, to a Tunisian-Italian family. He lived on a farm there until the end of World War II, then moved to Rome. After finishing high school, he was sent to the United States to live with relatives in Woyming and eventually made his way to California, becoming a student of Charles Loewner at Stanford in the early 1950s. Following his Ph.D., Adriano held positions at MIT, the University of Minnesota, and Caltech before joining the nascent mathematics department at the University of California, San Diego, in 1966 where he spent the remainder of his career. He passed away in San Diego on October 6, 2024, at the age of 96.

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arXiv2025

Schur--Weyl duality for diagonalizing a Markov chain on the hypercube

We show how the tools of modern algebraic combinatorics -- representation theory, Murphy elements, and particularly Schur--Weyl duality -- can be used to give an explicit orthonormal basis of eigenfunctions for a "curiously slowly mixing Markov chain" on the space of binary $n$-tuples. The basis is used to give sharp rates of convergence to stationarity.

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arXiv2025

A curiously slowly mixing Markov chain

We study a Markov chain with very different mixing rates depending on how mixing is measured. The chain is the "Burnside process on the hypercube $C_2^n$." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large $n$ is, in $\ell^1$ and in $\ell^2$. And started at general $x$, it mixes in at most $\log n$ steps in $\ell^1$. But, in $\ell^2$, it takes $\frac{n}{\log n}$ steps for most starting $x$. The $\ell^2$ mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors.

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arXiv2024

$c$-functions and Koornwinder polynomials

This paper develops the theory of Macdonald-Koornwinder polynomials in parallel analogy with the work done for the $GL_n$ case in [CR22]. In the context of the type $CC_n$ affine root system the Macdonald polynomials of other root systems of classical type are specializations of the Koornwinder polynomials. We derive $c$-function formulas for symmetrizers and use them to give $E$-expansions, principal specializations and norm formulas for bosonic, mesonic and fermionic Koornwinder polynomials. Finally, we explain the proof of the norm conjectures and constant term conjectures for the Koornwinder case.

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arXiv2024

Ian G. Macdonald: Works of Art

Ian Macdonald's works changed our perspective on so many parts of algebraic combinatorics and formal power series. This talk will display some selected works of the art of Ian Macdonald, representative of different periods of his oeuvre, and analyze how they resonate, both for the past development of our subject and for its future. This paper was prepared for the occasion of a lecture in tribute to Ian G. Macdonald, delivered at FPSAC 2024 in Bochum, Germany on 22 July 2024. I want to express thanks to the Executive Committee of FPSAC, the Organizing Committee of FPSAC 2024, and to the whole of our FPSAC 2024 community for making this lecture a possibility and for considering me for its delivery. Macdonald is my hero, and to be asked to play such a role in his legacy touches me deeply.

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arXiv2024

Lusztig varieties and Macdonald polynomials

This paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group $GL_n(\mathbb{F}_q)$. We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than $GL_n$. The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b) that traces on the Hecke algebra determine elements of the center of the Hecke algebra, (c) that the Geck-Rouquier basis elements of the center of the Hecke algebra produce an `expansion matrix', (d) that the parabolic subalgebras of the Hecke algebra produce a `contraction matrix' and (e) that the combination `expansion-contraction' is the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials.

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arXiv2023

Clebsch-Gordan coefficients for Macdonald polynomials

In this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products $E_\ell P_m$ and $P_\ell P_m$ for type $SL_2$ and type $GL_2$ Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from $2\cdot 3^{\ell-1}$ signed terms to $2\ell$ positive terms. We show that our rule for $P_\ell P_m$ is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing $E_\ell\mathbf{1}_0$ and $\mathbf{1}_0 E_\ell \mathbf{1}_0$ in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by $E_\ell$ and $P_\ell$. The formulas for a specific products $E_\ell P_m$ and $P_\ell P_m$ are obtained by evaluating the universal formulas at $t^{-\frac12}q^{-\frac{m}{2}}$.

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arXiv2022

Monk rules for type $GL_n$ Macdonald polynomials

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta (2008), but our method of derivation is to use Cherednik's interwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip (2010). Specializations of the Monk formula's at $q=0$ and/or $t=0$ provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials.

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arXiv2022

Set-valued tableaux for Macdonald polynomials

Set-valued tableaux formulas play an important role in Schubert calculus. Using the box greedy reduced word for the construction of the Macdonald polynomials, we convert the alcove walk formula for Macdonald polynomials to a set-valued tableaux formula for Macdonald polynomials. Our hope is that providing set-valued tableaux formulas for Macdonald polynomials will help to strengthen the analogies and possible connections between the calculus of Macdonald polynomials and Schubert calculus.

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arXiv2022

c-functions and Macdonald polynomials

This is a paper about $c$-functions and Macdonald polynomials. There are $c$-function formulas for $E$-expansions of $P_\lambda$ and $A_{\lambda+\rho}$, principal specializations of $P_\lambda$ and $E_\mu$, for Macdonald's constant term formulas, and for the norms of Macdonald polynomials. Most of these follow from the creation formulas for Macdonald polynomials, providing alternative proofs to several results from Macdonald (2003). In addition, we prove the Boson-Fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials.

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arXiv2022

Double Coset Markov Chains

Let $G$ be a finite group. Let $H, K$ be subgroups of $G$ and $H \backslash G / K$ the double coset space. Let $Q$ be a probability on $G$ which is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$). The random walk driven by $Q$ on $G$ projects to a Markov chain on $H \backslash G /K$. This allows analysis of the lumped chain using the representation theory of $G$. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $GL_n(q)$ onto a Markov chain on $S_n$ via the Bruhat decomposition. The chain on $S_n$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets. Some extensions and examples of double coset Markov chains with $G$ a compact group are discussed.

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arXiv2021

Comparing formulas for type $GL_n$ Macdonald polynomials: Supplement

This paper is a supplement to Guo-Ram arXiv:2104.02942, containing examples, remarks and additional material that could be useful to researchers working with Type $GL_n$ Macdonald polynomials. In the course of our comparison of the alcove walk formula and the nonattacking fillings formulas for type $GL_n$ Macdonald polynomials we did many examples and significant analysis of the literature. In the preparation of Guo-Ram arXiv:2104.02942 it seemed sensible to produce a document with focus and this material was removed. This is paper resurrects and organizes that material, in hopes that others may also find it useful.

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arXiv2021

Comparing formulas for type $GL_n$ Macdonald polynomials

The paper compares (and reproves) the alcove walk and the nonattacking fillings formulas for type $GL_n$ Macdonald polynomials which were given in Haglund-Haiman-Loehr arXiv:math.CO/0601693, Alexandersson arXiv:1602.05153 and Ram-Yip arXiv:0803.1146. The "compression" relating the two formulas in this paper is the same as that of Lenart arXiv:0804.4716. We have reformulated it so that it holds without conditions and so that the proofs of the alcove walk formula and the nonattacking fillings formula are parallel. This reformulation highlights the role of the double affine Hecke algebra and Cherednik's intertwiners. An exposition of the type $GL_n$ double affine braid group, double affine Hecke algebra, and all definitions and proofs regarding Macdonald polynomials are provided to make this paper self contained.

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arXiv2020

Calibrated representations of two boundary Temperley-Lieb algebras

The two boundary Temperley-Lieb algebra $TL_k$ arises in the transfer matrix formulation of lattice models in Statistical Mechanics, in particular in the introduction of integrable boundary terms to the six-vertex model. In this paper, we classify and study the calibrated representations---those for which all the Murphy elements (integrals) are simultaneously diagonalizable---which, in turn, corresponds to diagonalizing the transfer matrix in the associated model. Our approach is founded upon the realization of $TL_k$ as a quotient of the type $C_k$ affine Hecke algebra $H_k$. In previous work, we studied this Hecke algebra via its presentation by braid diagrams, tensor space operators, and related combinatorial constructions. That work is directly applied herein to give a combinatorial classification and construction of all irreducible calibrated $TL_k$-modules and explain how these modules also arise from a Schur-Weyl duality with the quantum group $U_q\mathfrak{gl}_2$.

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arXiv2019

Positive level, negative level and level zero

This is a survey on the combinatorics and geometry of integrable representations of quantum affine Lie algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal graphs and Macdonald polynomials that provide detailed understanding of the structure of the extremal weight modules and their characters. The final section surveys the alcove walk method of working with the positive level, negative level and level zero affine flag varieties and describes the corresponding actions of the affine Hecke algebra.

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arXiv2018

A Fock space model for decomposition numbers for quantum groups at roots of unity

In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module $\mathcal{F}_{\ell}$ with bar involution by specifying generators and "straightening relations" adapted from those appearing in the Kashiwara-Miwa-Stern formulation of the $q$-Fock space. By relating $\mathcal{F}_{\ell}$ to the corresponding affine Hecke algebra we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan-Lusztig polynomials. This property and the convenient combinatorial labeling of bases of $\mathcal{F}_{\ell}$ by dominant integral weights makes $\mathcal{F}_{\ell}$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.

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arXiv2018

The thickness of Schubert cells as incidence structures

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.

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