Related papers: Parking Spaces

Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of…

$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest…

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show…

In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval…

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be…

We study the structure and representation theory of the principal W-algebra $\mathsf{W}^{\mathsf{k}}_{\mathrm{pr}}$ of $\mathsf{V}^{\mathsf{k}}(\mathfrak{psl}_{2|2})$. The defining operator product expansions are computed, as is the Zhu…

We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\alpha_i)^{\lambda_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where…

We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices. We give a combinatorial…

In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H.…

Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the…

The lattice of intersections of reflecting hyperplanes of a complex reflection group W may be considered as the poset of 1-eigenspaces of the elements of W. In this paper we replace 1 with an arbitrary eigenvalue and study the topology and…

Let $W$ be the Weyl group corresponding to a finite dimensional simple Lie algebra $\mathfrak{g}$ of rank $\ell$ and let $m>1$ be an integer. In [I21], by applying cluster mutations, a $W$-action on $\mathcal{Y}_m$ was constructed. Here…

The action of a Weyl group on the associated weight lattice induces an additive action on the symmetric algebra and a multiplicative action on the group algebra of the lattice. We show that the coinvariant space of the multiplicative action…

Motivated by Lusztig's $G$-stable pieces, we consider the combinatorial pieces: the pairs $(w, K)$ for elements $w$ in the Weyl group and subsets $K$ of simple reflections that are normalized by $w$. We generalize the notion of cyclic shift…

We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the G-CW-version E_F(G) and the numerable G-space version J_F(G). They agree if G is discrete, or if G is a Lie group…

We study representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently "minimal" actions. When the group in question is PSL(2,R), we exhibit a one-one…

Motivated by the incidence problems between points and flats of a symplectic polar space, we study a large class of submodules of the space of functions on the standard module of a finite symplectic group of odd characteristic. Our…

Warped conformal field theories (WCFTs) are a novel class of non-relativistic theories. A simple, yet non-trivial, example of such theory is a massive Weyl fermion in $(1+1)$-dimensions, which we study in detail. We derive general…

Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If…

For a fixed integer $k$, we consider the set of noncrossing partitions, where both the block sizes and the difference between adjacent elements in a block is $1\bmod k$. We show that these $k$-indivisible noncrossing partitions can be…