Searcharxiv⌕ Search

arXiv subjects

Aritra Bhattacharya

Publications and source records attributed to Aritra Bhattacharya.

8 recordsLinked to original sources
arXiv2026

On factorization of matrix of Kazhdan-Lusztig polynomials

Let $\mathcal{H} = \mathcal{H}(W,S)$ be the Hecke algebra of the Coxeter system $(W,S)$ over $\mathbb{Z}[q^{\pm1}]$, where $W$ is the Weyl group of a symmetrizable Kac-Moody algebra. In this paper, we show that the matrix of Kazhdan-Lusztig polynomials of $\mathcal{H}$ factorizes into a product of $|S|$ many matrices, each of which has entries as polynomials in $q$ with nonnegative coefficients. To achieve this goal, we use hybrid basis $TC^J$ for $J\subseteq S$ of $\mathcal{H}$, defined by Grojnowski-Haiman. The intermediate matrices in the aforementioned factorization turn out to be the transition matrices from $TC^J$-basis to $TC^I$-basis for $I\subset J$. Equivalently, these coefficients can be computed using a natural restriction map from $\mathcal{H}$ to the parabolic Hecke algebra $\mathcal{H}_J$. Moreover, following the ideas from Grojnowski-Haiman, we also give a geometric proof of the positivity of these coefficients.

math.RT↗
arXiv2025

A tableaux formula for $q$-rook numbers

We provide a formula for the Garsia-Remmel $q$-rook numbers as a sum over standard Young tableaux. We connect our formula with the coefficients in $q$-Whittaker expansion of unicellular LLT functions.

math.CO↗
arXiv2024

$q$-Whittaker polynomials: bases, branching and direct limits

We study $q$-Whittaker polynomials and their monomial expansions given by the fermionic formula, the inv statistic of Haglund-Haiman-Loehr and the quinv statistic of Ayyer-Mandelshtam-Martin. The combinatorial models underlying these expansions are partition overlaid patterns and column strict fillings. The former model is closely tied to representations of the affine Lie algebra $\widehat{\mathfrak{sl}_n}$ and admits projections, branching maps and direct limits that mirror these structures in the Chari-Loktev basis of local Weyl modules. We formulate novel versions of these notions in the column strict fillings model and establish their main properties. We construct weight-preserving bijections between the models which are compatible with projection, branching and direct limits. We also establish connections to the coloured lattice paths formalism for $q$-Whittaker polynomials due to Wheeler and collaborators.

math.CO↗
arXiv2024

The monomial expansion formula for Hall-Littlewood $P$-polynomials

We give a Hecke algebra derivation of Macdonald's expansion formula for Hall-Littlewood polynomials in terms of semistandard Young tableaux. This is accomplished by first obtaining a Hecke algebra lift of the expansion coefficients and then proving a generalization of Klostermann's recursions.

math.CO↗
arXiv2023

Monomial expansions for $q$-Whittaker and modified Hall-Littlewood polynomials

We consider the monomial expansion of the $q$-Whittaker polynomials given by the fermionic formula and via the inv and quinv statistics. We construct bijections between the parametrizing sets of these three models which preserve the $x$- and $q$-weights, and which are compatible with natural projection and branching maps. We apply this to the limit construction of local Weyl modules and obtain a new character formula for the basic representation of $\widehat{\mathfrak{sl}_n}$. Finally, we indicate how our main results generalize to the modified Hall-Littlewood case.

math.CO↗
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 $t^{-\frac12}q^{-\frac{m}{2}}$.

math.RT↗
arXiv2022

Haglund's positivity conjecture for multiplicity one pairs

Haglund's conjecture states that $\dfrac{\langle J_λ(q,q^k),s_μ\rangle}{(1-q)^{|λ|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $λ,μ$ and all non-negative integers $k$, where $J_λ$ is the integral form Macdonald symmetric function and $s_μ$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(λ,μ)$ satisfies $K_{λ,μ}=1$ or $K_{μ',λ'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.

math.CO↗