Related papers: Ideals and quotients of B-quasisymmetric functions

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that any set of finite perimeter can be approximated in the BV norm by a set whose topological and measure theoretic boundaries…

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping…

The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular…

We explain how to use representation theory to give a lower bound on the dimension of the quotient ring by type $B_n$ diagonal invariants that improves upon the current known lower bound $(2n+1)^n$ by a quadratic polynomial in $n$.

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \ne A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given…

The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to…

Guo and the second author have shown that the closure $[I]$ in the Drury-Arveson space of a homogeneous principal ideal $I$ in $\mathbb{C}[z_1,...,z_n]$ is essentially normal. In this note, the authors extend this result to the closure of…

Approximate duals of B-splines were first used by Chui et al. (2004) for the purpose of constructing tight wavelet frames on bounded intervals. They are splines with local support, whose inner product with a polynomial in the spline space…

We study the symbolic powers of the Stanley-Reisner ideal $I_{B_n}$ of a bipyramid $B_n$ over a $n-$gon $Q_n$. Using a combinatorial approach, based on analysis of subtrees in $Q_n$ we compute the Waldschmidt constant of $I_{B_n}$.

We construct a complex of toric varieties we call the quasisymmetric Grassmannian inside the Grassmannian of $r$-planes in $\mathbb{C}^n$. Each irreducible component is a positroid variety and an $S_n$ translate of a toric Richardson…

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is…

Given two finite sequences of positive integers $\alpha$ and $\beta$, we associate a square free monomial ideal $I_{\alpha,\beta}$ in a ring of polynomials $S$, and we recursively compute the algebraic invariants of $S/I_{\alpha,\beta}$.…

We introduce the notion of a polygonic spectrum which is designed to axiomatize the structure on topological Hochschild homology $\mathrm{THH}(R,M)$ of an $\mathbb{E}_1$-ring $R$ with coefficients in an $R$-bimodule $M$. For every polygonic…

In the context of generating functions for $P$-partitions, we revisit three flavors of quasisymmetric functions: Gessel's quasisymmetric functions, Chow's type B quasisymmetric functions, and Poirier's signed quasisymmetric functions. In…

We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…

The Schur functions, a basis for the symmetric polynomials (Sym), encode the irreducible representations of the symmetric group, $\mathfrak{S}_n$, via the Frobenius characteristic map. In 1996, Krob and Thibon defined a quasisymmetric…

We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the…

Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial…

Schubert polynomials are distinguished representatives of Schubert cycles in the cohomology of the flag variety. In the spirit of Bergeron and Sottile, we use the Bruhat order to give $(n-1)!$ different combinatorial formulas for the…

We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and (q_{k,n})_k related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the…