Quantum and affine Schubert calculus and Macdonald polynomials

Abstract

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-A affine Weyl group. We construct two one-parameter families of functions that respectively transition positively with Hall-Littlewood and Macdonald's P-functions, and specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. Our approach leads us to conjecture that all elements in a defining set of 3-point genus 0 Gromov-Witten invariants for flag manifolds can be formulated as strong covers.Assuming this conjecture, since A (k) λ (x; t) is a t-positive sum of Schur functions, the Macdonald/Schur transition matrices factor over N [q, t]. The construction of A (k) µ (x; t) is extremely intricate and the conjectures remain unproven as a consequence. Nevertheless, their study inspired discoveries in representation theory [Hai08] and suggested a connection between the theory of Macdonald polynomials and quantum and affine Schubert calculus.The affine Grassmannian of G = S L(n, C) is given by Gr] is the ring of formal power series and C((t)) = C[[t]][t −1 ] is the ring of formal Laurent series. Quillen (unpublished) and Garland and Raghunathan [GR75] showed that Gr is homotopy-equivalent to the group Ω SU(n, C) of based loops into SU(n, C). The homology H * (Gr) and cohomology H * (Gr) thus have dual Hopf algebra structures which, using results of [Bot58], can be explicitly identified by a subring Λ (n) and a quotient Λ (n) of Λ.On one hand, the algebraic nil-Hecke ring construction of Kostant and Kumar [KK86] and the work of Peterson [Pet97] developed the study of Schubert bases associated to Schubert cells of Gr in the Bruhat decomposition of G (C((t))),indexed by Grassmannian elements of the affine Weyl groupà n−1 . On the other, inspired by an empirical study of the polynomials A (k) λ (x; t) when t = 1, distinguished bases for Λ (n) and Λ (n) that refine the Schur basis for Λ were introduced and connected to the quantum cohomology of Grassmannians in [LM08, LM07]. The two approaches merged when Lam proved in [Lam08] that these k-Schur bases are sets of representatives for the Schubert classes of H * (Gr) and H * (Gr) (where k = n − 1). Moreover, the k-Schur functions for Λ (k) = Λ t=1 (k) were conjectured [LM05] to be the parameterless {A (k) λ (x; 1)}, suggesting a link from the theory of Macdonald polynomials to quantum and affine Schubert calculus.Here, we circumvent the problem that the characterization for A (k) µ (x; t) lacks in mechanism for proof and definitively establish this link. Our work relies on a remarkable connection between chains in the strong and the weak order poset on the type-A affine Weyl group. From this, we are able to construct one parameter families of symmetric functions that transition positively with H µ (x; 0, t) and Macdonald's P-functions and that specialize to the Schubert repres...

Closed 𝑘-Schur Katalan functions as 𝐾-homology Schubert representatives of the affine Grassmannian

“…Although we know a dictionary between quantum and affine Schubert classes, the problem of translating the Chevalley formula into its affine counterpart seems not to be so simple. For the homology case, the analogous issue was pursued by Dalal and Morse [9,Conjecture 39]. Another basic question is what formula on the quantum side is corresponding to g(k…”

Positivity of affine charge

“…The intense study [AB12] of covers in the Bruhat order on the affine symmetric group sheds light on the conjectured characterization for k-atoms [LLMS10] as generating functions for marked saturated chains. The work of [DM15] introduces a notion of affine charge on affine Bruhat counter-tableaux, objects in bijection with k-tableaux, and definitively proves that Macdonald polynomials and quantum and affine Schubert calculus are interconnected.…”

Hopf dreams and diagonal harmonics

“…The discovery of these polynomials originated important developments including the proof of the Macdonald constant-term identities [25] and the resolution of the Macdonald positivity conjecture [42], as well as many results in connection with representation theory of quantum groups [27], affine Hecke algebras [45,46,56], and the Calogero-Sutherland model in particle physics [50]. Diagonal harmonics is also connected to many areas in mathematics where they play a central role, including the rectangular and rational Catalan combinatorics in algebraic combinatorics [5,6,9,13,59], cohomology of flag manifolds and group schemes in algebraic topology [26], Hilbert schemes in algebraic geometry [42,43], homology of torus links in knot theory [36,60], and more.…”