We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies
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Abstract of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann.As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial.In particular, we obtain a formula for$\mathrm{\Lambda}({X}^{m,n})\subset \mathcal{E}$ \Lambda(X^{m{,}n})\subset\mathcal{E} which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one.${\nabla}^{m}{s}_{\lambda}$ \nabla^{m}s_{\lambda} Free, publiclyaccessible full text available April 23, 2025 
Tauman Kalai, Yael (Ed.)In 2003, Cohn and Umans proposed a grouptheoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining ω = 2, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored. We first show that groups of Lie type cannot prove ω = 2 within the grouptheoretic approach. This is based on a representationtheoretic argument that identifies the secondsmallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers' result concerning productfree sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being selfnormalizing. Our barrier results leave open several natural paths to obtain ω = 2 via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of ω = 2 in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving ω = 2. We give two constructions in the continuous setting, each of which evades one of our two barriers.more » « less

Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.more » « less

Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.more » « less

We prove that the KkSchur functions are part of a family of inhomogenous symmetric functions whose top homogeneous components are Catalan functions, the Euler characteristics of certain vector bundles on the flag variety. LamSchillingShimozono identified the KkSchur functions as Schubert representatives for Khomology of the affine Grassmannian for SL_{k+1}. Our perspective reveals that the KkSchur functions satisfy a shift invariance property, and we deduce positivity of their branching coefficients from a positivity result of Baldwin and Kumar. We further show that a slight adjustment of our formulation for KkSchur functions produces a second shiftinvariant basis which conjecturally has both positive branching and a rectangle factorization property. Building on work of IkedaIwaoMaeno, we conjecture that this second basis gives the images of the LenartMaeno quantum Grothendieck polynomials under a Ktheoretic analog of the Peterson isomorphism.more » « less