Related papers: Rook placements and orbit harmonics

For fixed positive integers $n,m,r$, let $\mathrm{Mat}_{n \times m}(\mathbb{C})$ be the affine space of $n \times m$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times m}]$. We define a homogeneous ideal $I_{n,m,r}$,…

Let $\mathcal{M}_{n,a}$ be the set consisting of involutions in symmetric group $\mathfrak{S}_n$ with exactly $a$ fixed points and apply the orbit harmonics method to obtain a graded $\mathfrak{S}_n$-module $R(\mathcal{M}_{n,a})$. Liu, Ma,…

Let $\mathrm{Mat}_{n \times n}(\mathbb{C})$ be the affine space of $n \times n$ complex matrices with coordinate ring $\mathbb{C}[\mathbf{x}_{n \times n}]$. We define graded quotients of $\mathbb{C}[\mathbf{x}_{n \times n}]$ which carry an…

We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$…

A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic…

Let $G$ be a complex reductive group, $B$ be a Borel subgroup of G, $\nt$ be the Lie algebra of the unipotent radical of $B$, and $\nt^*$ be its dual space. Let $\Phi$ be the root system of $G$, and $\Phi^+$ be the set of positive roots…

In 1986 Harer and Zagier computed a certain matrix integral to determine an influential closed-form formula for the number of (orientable) one-face maps on n vertices colored from N colors. Kerov (1997) provided a proof which computed the…

We prove an asymptotic formula for the number of fixed rank matrices with integer coefficients over a number field K/Q and bounded norm. As an application, we derive an approximate Rogers integral formula for discrete sets of module…

Let $\Pi_{(b^a)}$ be the locus of unordered set partitions of $[ab]$ with $a$ blocks of size $b$. We embed unordered set partitions of $[n]$ into the affine space $\mathbb{C}^{\binom{[n]}{2}}$ with coordinate ring…

We study Schroder paths drawn in a (m,n) rectangle, for any positive integers m and n. We get explicit enumeration formulas, closely linked to those for the corresponding (m,n)-Dyck paths. Moreover we study a Schroder version of…

For each certain "nice" piecewise linear function $f:[0,1] \to [0,1]$, we consider a family of growing Young diagrams $\{\lambda(f,N)\}_{N=1}^{\infty}$ by enlarging the region under the graph of $f$. We compute asymptotic formulas for the…

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective…

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three…

We consider $m \times s$ matrices (with $m\geq s$) in a real affine subspace of dimension $n$. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is…

We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is…

We investigate harmonic maps in the context of isometric embeddings when the target space is Ricci-flat and has codimension one. With the help of the Campbell-Magaard theorem we show that any $n$-dimensional ($n\geqslant 3$) Lorentzian…

We study a concrete family of symmetric integral $Z$-matrices attached to weighted star trees. The arms are ordinary type-$A$ chains and the central diagonal entry is an arbitrary positive integer $k$ rather than being fixed to the Cartan…

Let $U$ be a maximal unipotent subgroup in a symplectic group over a finite field of sufficiently large characteristic $p$. According to the Kirillov's orbit method, the coadjoint orbits of the group $U$ play the key role in the description…

Study of the matrix-level affine algebra $U_{m,K}$ is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of $U_{m,K}$ modular-invariant partition functions. Here we connect the…