Christopher Seaton
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View article: When does the zero fiber of the moment map have rational singularities?
When does the zero fiber of the moment map have rational singularities? Open
Let $G$ be a complex reductive group and $V$ a $G$-module. There is a natural moment mapping $\mu\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote $\mu^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results of Musta\c{t}\u{a} …
View article: Hilbert measures on orbit spaces of coregular $\operatorname{O}_m$-modules
Hilbert measures on orbit spaces of coregular $\operatorname{O}_m$-modules Open
We construct canonical measures, referred to as Hilbert measures, on orbit spaces of classical coregular representations of the orthogonal groups $\operatorname{O}_m$. We observe that the measures have singularities along non-principal str…
View article: Euler characteristics of linear symplectic quotients and $\operatorname{O}(2)$-spaces
Euler characteristics of linear symplectic quotients and $\operatorname{O}(2)$-spaces Open
We give explicit computations of the $Γ$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the circle…
View article: A symmetric function approach to polynomial regression
A symmetric function approach to polynomial regression Open
We give an explicit solution formula for the polynomial regression problem in terms of Schur polynomials and Vandermonde determinants. We thereby generalize the work of Chang, Deng, and Floater to the case of model functions of the form $\…
View article: A universal Euler characteristic of non-orbifold groupoids and Riemannian structures on Lie groupoids
A universal Euler characteristic of non-orbifold groupoids and Riemannian structures on Lie groupoids Open
We introduce the universal Euler characteristic of an orbit space definable groupoid, a class of groupoids containing cocompact proper Lie groupoids as well as translation groupoids for semialgebraic group actions. Generalizing results of …
View article: Isomorphisms of Symplectic Torus Quotients
Isomorphisms of Symplectic Torus Quotients Open
We call a reductive complex group $G$ quasi-toral if $G^0$ is a torus. Let $G$ be quasi-toral and let $V$ be a faithful $1$-modular $G$-module. Let $N$ (the shell) be the zero fiber of the canonical moment mapping $μ\colon V\oplus V^*\to\m…
View article: Higher Koszul Brackets on the Cotangent Complex
Higher Koszul Brackets on the Cotangent Complex Open
Let $A=\boldsymbol{k}[x_1,x_2,\dots ,x_n]/I$ be a commutative algebra where $\boldsymbol{k}$ is a field, $\operatorname{char}(\boldsymbol{k})=0$, and $I\subseteq S:=\boldsymbol{k}[x_1,x_2,\dots , x_n]$ a Poisson ideal. It is well known tha…
View article: Orbifold Euler characteristics of non‐orbifold groupoids
Orbifold Euler characteristics of non‐orbifold groupoids Open
For a finitely presented discrete group $\\Gamma$, we introduce two\ngeneralizations of the orbifold Euler characteristic and $\\Gamma$-orbifold\nEuler characteristic to a class of proper topological groupoids large enough to\ninclude all …
View article: Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups
Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups Open
We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group $\operatorname{O}_2$. The multigradings considered include the maximal grading associated to the decomp…
View article: When does the zero fiber of the moment map have rational singularities?
When does the zero fiber of the moment map have rational singularities? Open
Let $G$ be a complex reductive group and $V$ a $G$-module. There is a natural moment mapping $μ\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote $μ^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results of Mustaţă [Mus01] to f…
View article: The spectra of digraphs with Morita equivalent $C^\ast$-algebras
The spectra of digraphs with Morita equivalent $C^\ast$-algebras Open
Eilers et al. have recently completed the geometric classification of unital graph $C^\ast$-algebras up to Morita equivalence using a set of moves on the corresponding digraphs. We explore the question of whether these moves preserve the n…
View article: Orbifold Euler characteristics of non-orbifold groupoids
Orbifold Euler characteristics of non-orbifold groupoids Open
For a finitely presented discrete group $Γ$, we introduce two generalizations of the orbifold Euler characteristic and $Γ$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all cocompact prope…
View article: The Hilbert series of SL2-invariants
The Hilbert series of SL2-invariants Open
Let [Formula: see text] be a finite-dimensional representation of the group [Formula: see text] of [Formula: see text] matrices with complex coefficients and determinant one. Let [Formula: see text] be the algebra of [Formula: see text]-in…
View article: Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$
Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$ Open
We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an $\operatorname{SU}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansi…
View article: An algebra generated by $x - 2$
An algebra generated by $x - 2$ Open
By a theorem of R. Stanley, a graded Cohen-Macaulay domain $A$ is Gorenstein if and only if its Hilbert series satisfies the functional equation \[
\operatorname{Hilb}_A(t^{-1})=(-1)^d t^{-a}\operatorname{Hilb}_A(t), \] where $d$ is the …
View article: Symplectic reduction at zero angular momentum
Symplectic reduction at zero angular momentum Open
We study the symplectic reduction of the phase space describing $k$ particles\nin $\\mathbb{R}^n$ with total angular momentum zero. This corresponds to the\nsingular symplectic quotient associated to the diagonal action of\n$\\operatorname…
View article: Differentiable stratified groupoids and a de Rham theorem for inertia spaces
Differentiable stratified groupoids and a de Rham theorem for inertia spaces Open
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively different…
View article: On compositions with 𝑥²/(1-𝑥)
On compositions with 𝑥²/(1-𝑥) Open
In the past, empirical evidence has been presented that Hilbert series of symplectic quotients of unitary representations obey a certain universal system of infinitely many constraints. Formal series with this property have been called sym…