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Over a field of characteristic two, we develop a theory of standard monomials for polynomial rings modulo a Frobenius power of the maximal ideal generated by all variables. As a result, we obtain a filtration by modular GL_n-representations…

Let $R$ be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial…

We prove a conjecture of Rouquier relating the decomposition numbers in category $\mathcal{O}$ for a cyclotomic rational Cherednik algebra to Uglov's canonical basis of a higher level Fock space. Independent proofs of this conjecture have…

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…

We extend the Razumov-Stroganov conjecture relating the groundstate of the O(1) spin chain to alternating sign matrices, by relating the groundstate of the monodromy matrix of the O(1) model to the so-called refined alternating sign…

We study the description of the $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory in terms of the modes of the spin-1/2 affine primary field $\phi^\alpha$. These are shown to satisfy generalized `canonical commutation…

We prove a conjecture of Drake and Kim: the number of $2$-distant noncrossing partitions of $\{1,2,...,n\}$ is equal to the sum of weights of Motzkin paths of length $n$, where the weight of a Motzkin path is a product of certain fractions…

We introduce a fermionic formula associated with any quantum affine algebra U_q(X^{(r)}_N). Guided by the interplay between corner transfer matrix and Bethe ansatz in solvable lattice models, we study several aspects related to…

Let $X=(x_{ij})$ and $Y=(y_{ij})$ be generic $n$ by $n$ matrices and $Z=XY-YX$. Let $S=k[x_{11},...,x_{nn},y_{11},...,y_{nn}]$, where $k$ is a field, let $I$ be the ideal generated by the entries of $Z$ and let $R=S/I$. We give a conjecture…

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family $R_k$, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We…

We prove a fermionic-bosonic duality relation for the Macdonald index in Argyres-Douglas theories of type $(A_1, D_{2k+1})$, thereby yielding a conjectural fermionic formula due to Andrews et al. Our duality is built upon a new conjugate…

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess-Zumino-Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and…

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov's Heisenberg category H…

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative dagger-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous…

Let $r$ be a positive integer and let $G_n$ be the reflection group of $n \times n$ monomial matrices whose entries are $r^{th}$ complex roots of unity and let $k \leq n$. We define and study two new graded quotients $R_{n,k}$ and $S_{n,k}$…

In the present paper by Frobenius algebra Y we mean a finite dimensional algebra possessing an associative and invertible (nondegenerate) form a scalar product, referred to as the Frobenius structure. The nondegenerate form has an inverse.…

In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric…

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible…

Let $F$ be a field. Following the resolution of Milnor's conjecture relating the graded Witt ring of $F$ to its mod-2 Milnor $K$-theory, a major problem in the theory of symmetric bilinear forms is to understand, for any positive integer…

We introduce a variety $Y_{n,k}$, which we call the \textit{affine $\Delta$-Springer fiber}, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an $S_n$ action and a bigrading that corresponds to the…