arXiv subjects
Publications and source records attributed to Claire Frechette.
We introduce a method for producing vector-valued automorphic forms on unitary groups from scalar-valued ones. As an application, we construct an explicit example. Our strategy employs certain differential operators. It is inspired by work of Cl\'ery and van der Geer in the setting of Siegel modular forms, but it also requires overcoming challenges that do not arise in the Siegel setting.
Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums associated to their products. More precisely, we let $f_j$, $j=1,2$ be such that $|f_j(n)|\leq τ(n)^κ$ for some $κ\in\mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\geq ηx_j (\log x_j)^{2^κ-1}$ for some $x_1, x_2\gg 1$ and $η>\max_j\{(\log x_j)^{-1/100}\}$. We then show that there exists $x\geq \min\{x_1, x_2\}^{ξ^2}$ such that $\left|S(x,f_1f_2)\right|\geq ξx (\log x)^{2^{2κ}-1}$, where $ξ=Cη^{1+2^{κ+3}}$ for some absolute constant $C>0$.
Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} λ_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq x}λ_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq k^ε$. Lamzouri proved in arXiv:1703.10582 [math.NT] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $ε>(\log k)^{-\frac{1}{8}}$ and $1\leq T\leq (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\geq k^ε$ provided that $L(s,f)$ has no more than $ε^2\log k/5000$ zeros in the region $\left\{s\,:\, \Re(s)\geq \frac34, \, |\Im(s)-ϕ| \leq \frac14\right\}$ for every real number $ϕ$ with $|ϕ|\leq T$.
Whittaker functions are special functions that arise in $p$-adic number theory and representation theory. They may be defined on representations of reductive groups as well as their metaplectic covering groups: fascinatingly, many of their number theoretic applications survive the transition between the reductive and metaplectic cases. However, one notable difference is that the space of Whittaker functions on a reductive group over a nonarchimedean local field $F$ is one-dimensional, whereas this is no longer true in the metaplectic case. In a previous paper, the second author showed that the dimension of the space of Whittaker functions on an arbitrary $n$-fold metaplectic cover of $GL_r(F)$ can be counted in terms of the number of solutions to a particular system of linear Diophantine equations in terms of $n$ and $r$. In this paper, we calculate two precise formulae for $\dim(\mathfrak{W})$, one inspired by viewing this system as a homogenous specialization of an inhomogenous system and the other by the structure of the coroot lattice of $GL_r(F)$. Then we use these formulae to investigate a homomorphism between $\mathfrak{W}$ and a particular quantum group module, built by the second author in a previous paper, and show precisely when this map is well-defined for any choice of basis for $\mathfrak{W}$.
We introduce a solvable lattice model for supersymmetric LLT polynomials, also known as super LLT polynomials, based upon particle interactions in super n-ribbon tableaux. Using operators on a Fock space, we prove a Cauchy identity for super LLT polynomials, simultaneously generalizing the Cauchy and dual Cauchy identities for LLT polynomials. Lastly, we construct a solvable semi-infinite Cauchy lattice model with a surprising Yang-Baxter equation and examine its connections to the Cauchy identity.
We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(β,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $β$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $n$ pairs of variables is a Drinfeld twist of the $U_q(\widehat{\mathfrak{sl}}_{n+1})$ $R$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $β$-Grothendieck polynomials and dual double $β$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $β$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $β$-Grothendieck polynomials, and prove a new branching rule for double $β$-Grothendieck polynomials.
In this paper, we extend results connecting quantum groups to spherical Whittaker functions on metaplectic covers of $GL_r(F)$, for $F$ a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic $n$-fold cover of $GL_r(F)$ a set of Yang-Baxter equations model the action of standard intertwiners on principal series Whittaker functions. These equations arise from a Drinfeld twist of the quantum affine Lie superalgebra $U_{\sqrt{v}}(\widehat{\frak{gl}}(n)),$ where $v = q^{-1}$ for $q$ the cardinality of the residue field. We extend their results to all metaplectic covers of $GL_r(F)$, providing new solutions to Yang-Baxter equations matching the scattering matrix for the associated Whittaker functions. Each cover has an associated integer invariant $n_Q$ and the resulting solutions are connected to the quantum group $U_{\sqrt{v}}(\widehat{\frak{gl}}(n_Q))$ and quantum superalgebra $U_{\sqrt{v}}(\widehat{\frak{gl}}(1|n_Q))$.
In this paper, we study the parallel cases of Zagier's and Folsom-Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their $p$-adic properties under the action of Hecke operators.
Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_ν(a,b;q)=\sum_{\substack{λ\\[1pt] λ_1\leq a}} q^{c|λ|} P_{2λ}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}