A proof of the fermionic theta coinvariant conjecture

For the quantum cohomology of Grassmannians, see [1]. Given integers l, N with 1 6 l 0〈σλ, σμ, σν∨〉d q σν , where ν ∨ is the partition dual to ν. Let ζ be a primitive Nth root of (−1). Put K = Q(ζ), k = N − l and let Λ = Z[e1, e2, . . . ] be the (graded) ring of symmetric functions, where ei is the ith elementary symmetric function. Let hi be the ith complete symmetric function. Putting E(t) = P eit , H(t) = P hit , we have E(−t)H(t) = 1. The result of applying a symmetric function σ to a tuple (x1, . . . , xn) of arguments will be denoted by σ(x1, . . . , xn). Finally, we put Λ ′ = Λ/(el+1, el+2, . . . ), ΛQ = Λ ⊗ Q, ΛQ = Λ ′ ⊗ Q and ΛK = ΛQ ⊗Q K. Theorem 1 (Siebert–Tian [2]). The homomorphism of rings ST: Λ[q]→QH(G,Z) defined by ST (q) = 1 ⊗ q, ST (ei) = ci(S) ⊗ 1 is an epimorphism, and KerST = (el+1, el+2, . . . ;hN−l+1, . . . , hN−1, hN + (−1)q). Let QH(G,Q) = QH(G,Z) ⊗ Q be the rational quantum cohomology ring and let QH(G,Q, 1) denote its specialization for q = 1. Putting I1 = (hN−l+1, . . . , hN−1, hN + (−1)),

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

A result of Farahat and Higman shows that there is a “universal” algebra, FH , interpolating the centres of symmetric group algebras, Z ( ℤ S n ) . We explain that this algebra is isomorphic to ℛ ⊗ Λ , where ℛ is the ring of integer-valued polynomials and Λ is the ring of symmetric functions. Moreover, the isomorphism is via “evaluation at Jucys–Murphy elements”, which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products Γ ≀ S n of a fixed finite group Γ . This involves constructing wreath-product versions ℛ Γ and Λ ( Γ * ) of ℛ and Λ , respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, FH Γ , is isomorphic to ℛ Γ ⊗ Λ ( Γ * ) and use this to compute the p -blocks of wreath products.

Skew Weyl Modules for GLn and Degree Reduction for Schur Algebras

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.

Powers of staircase Schur functions and symmetric analogues of Bessel polynomials

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald’s constant term identities admit an extra set of free parameters, thereby linking them to Poincaré polynomials. We then exploit these extra degrees of freedom in the case of typeA\mathrm {A}to give the first proof of Kadell’s orthogonality conjecture—a symmetric function generalisation of theqq-Dyson conjecture or Zeilberger–Bressoud theorem.Key ingredients in our proof of Kadell’s orthogonality conjecture are multivariable Lagrange interpolation, the scalar product for Demazure characters and(0,1)(0,1)-matrices.

Double Kostka polynomials $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ are polynomials in $t$, indexed by double partitions $\boldsymbol{\lambda}, \boldsymbol{\mu}$. As in the ordinary case, $K_{\boldsymbol{\lambda}, \boldsymbol{\mu}}(t)$ is defined in terms of Schur functions $s_{\boldsymbol{\lambda}}(x)$ and Hall--Littlewood functions $P_{\boldsymbol{\mu}}(x;t)$. In this paper, we study combinatorial properties of $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ and $P_{\boldsymbol{\mu}}(x;t)$. In particular, we show that the Lascoux--Sch\"utzenberger type formula holds for $K_{\boldsymbol{\lambda},\boldsymbol{\mu}}(t)$ in the case where $\boldsymbol{\mu} = (-,\mu'')$. Moreover, we show that the Hall bimodule $\mathscr{M}$ introduced by Finkelberg-Ginzburg-Travkin is isomorphic to the ring of symmetric functions (with two types of variables) and the natural basis $\mathfrak{u}_{\boldsymbol{\lambda}}$ of $\mathscr{M}$ is sent to $P_{\boldsymbol{\lambda}}(x;t)$ (up to scalar) under this isomorphism. This gives an alternate approach for their result.