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This is the second in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the "Verlinde ring" of its loop group. We introduce the Dirac family of Fredholm operators…

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics…

We investigate actions of cyclic groups on polynomial rings with two variables, and the minimal free resolution of the corresponding invariant ring. In particular, we fully classify several cases, including the case the defining ideal has…

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean…

We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of $C_2$, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside…

We give a presentation of the $\mathrm{GL}_n(\mathbb{C})$-equivariant cohomology ring with $\mathbb{Z}$-coefficients of the variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C})) \subseteq \mathrm{GL}_n(\mathbb{C})^2$ for any $n$. It…

This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a…

The complete forms of the S-matrix elements of a transverse scalar field, two world volume gauge fields, and a Potential $C_{n-1}$ Ramond-Ramond (RR) form field are investigated. In order to find an infinite number of $t,s, (t+s+u)$-channel…

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group…

This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel…

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining…

It is well-known that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the Yangian in type $A$ yields generators of its Drinfeld presentation. Defining relations between these generators are known in an…

A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, and a presentation of the category is given; specifically, we prove that seven relations among its four generating homomorphisms suffice to deduce all…

Inspired by quantum information theory, we look for representations of the braid groups $B_n$ on $V^{\otimes (n+m-2)}$ for some fixed vector space $V$ such that each braid generator $\sigma_i, i=1,...,n-1,$ acts on $m$ consecutive tensor…

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for…

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula…

We prove the basic properties of determinantal semi-invariants for presentation spaces over any finite dimensional hereditary algebra over any field. These include the virtual generic decomposition theorem, stability theorem and the…

Let $C_2$ denote the cyclic group of order two. Given a manifold with a $C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded cohomology. In this paper, we develop a theory of fundamental classes for equivariant submanifolds…

In commutative invariant theory, a classical result due to Auslander says that if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is…

Our starting point is Mumford's conjecture, on representations of Chevalley groups over fields, as it is phrased in the preface of "Geometric Invariant Theory". After extending the conjecture appropriately, we show that it holds over an…