Related papers: A residue theorem for rational trigonometric sums …
The total multiplicity in the decomposition into irreducibles of the tensor product i x j of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them sum_k N_{i j}^{k}= sum_k N_{ibar j}^{k}. This…
Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic, but generally not rational, function of…
We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as…
Blowing up a rational surface singularity in a reflexive module gives a (any) partial resolution dominated by the minimal resolution. The main theorem shows how deformations of the pair (singularity, module) relates to deformations of the…
This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars.…
We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups.
We present a simplified formulation of open intersection numbers, as an alternative to the theory initiated by Pandharipande, Solomon and Tessler. The relevant moduli spaces consist of Riemann surfaces (either with or without boundary) with…
The main result describes the Brauer-Nesbitt reduction of unipotent representations of a finite group of Lie type, expressing it as an explicit linear combination of the restriction of Weyl modules from the algebraic group to the group of…
We formulate a conjectural Lefschetz formula for locally symmetric spaces of finite volume. The formula can be verified in the compact case and for Riemann surfaces.
We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g$ ranges over the exceptional complex simple Lie algebras. We give…
In this paper, we study sum formulas for Schur multiple zeta values and give a generalization of the sum formulas for multiple zeta(-star) values. We show that for ribbons of certain types, the sum over all admissible Young tableaux of this…
We construct an explicit and calculable models for rational U(2)-spectra. This is obtained by assembling seven blocks obtained in previous work: the toral part and earlier work on small toral groups. The assembly process requires detailed…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
The purpose of this paper is to prove dimension formulas for $T^1$ and $T^2$ for rational surface singularities. These modules play an important role in the deformation theory of isolated singularities in analytic and algebraic geometry.…
We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of…
We establish a structure theorem for rational maps $f:\overline{\mathbb{C}}\to\overline{\mathbb{C}}$: the pullback metric $f^{*}{\rm d}s_{0}^{2}$ of the standard metric ${\rm d}s_{0}^{2}$ admits a canonical decomposition into finitely many…
A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a…
We present bases for certain spaces of meromorphic vector-valued rational-weight mock modular forms constructed using Rademacher sums.
Moduli spaces of admissible covers and stable maps of target curves give rise to cycles on $\overline{M}_{g,n}$. We prove a formula relating these cycles. It recovers both the Ekedahl-Lando-Shapiro-Vainshtein formula and the…
The main goal of this article is to provide a proof of the Pederson-Roy-Szpirglas theorem about counting common real zeros of real polynomial equations by using basic results from Linear algebra and Commutative algebra. The main tools are…