Related papers: Combinatorics and Representation Theory for Genera…

We introduce the framework of general probabilistic theories (GPTs for short). GPTs are a class of operational theories that generalize both finite-dimensional classical and quantum theory, but they also include other, more exotic theories,…

We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order (p-primary) orbifold Euler characteristic of symmetric…

This is a chapter in an upcoming Tamari Festscrift. Permutahedra are a class of convex polytopes arising naturally from the study of finite reflection groups, while generalized associahedra are a class of polytopes indexed by finite…

Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$ Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a…

We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act…

We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies…

Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…

Let $G$ be a countable group, $\operatorname{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chabauty topology and $\operatorname{Is}(G) \subset \operatorname{Sub}(G)$ the collection of isolated points. We denote by…

Group field theories are a generalization of matrix models which provide both a second quantized reformulation of loop quantum gravity as well as generating functions for spin foam models. While states in canonical loop quantum gravity, in…

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra,…

Minkowski sums of simplices in ${\mathbb{R}}^n$ form an interesting class of polytopes that seem to emerge in various situations. In this paper we discuss the Minkowski sum of the simplices $\Delta_{k-1}$ in ${\mathbb{R}}^n$ where $k$ and…

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on…

In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain…

This paper generalizes a theorem of Hida on the structure of ordinary representations on unitary groups to $P$-ordinary representations, where $P$ is a general parabolic subgroup of some general linear group. When $P$ is minimal, we recover…

Unified field theories act to merge the internal symmetries of the standard model into a single group. Here we lay out something different. That is, instead of aiming to unify the internal symmetries, we demonstrate a sense in which the…

Through the glasses of didactic reduction: We consider a (periodic) tessellation $\Delta$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $\Delta$ we mean the process of cutting $M$ along corank-1…

Let $G$ be a transitive permutation group on $\Omega$. The $G$-invariant partitions form a sublattice of the lattice of all partitions of $\Omega$, having the further property that all its elements are uniform (that is, have all parts of…

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for…

For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such…

We describe the structure of the Whittaker or Gelfand-Graev module on a $n$-fold metaplectic cover of a $p$-adic group $G$ at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum…