arXiv subjects

Hans-Christian Herbig

Publications and source records attributed to Hans-Christian Herbig.

At least 19 recordsLinked to original sources

arXiv2026

The partial derivative of ratios of Schur polynomials and applications to symplectic quotients

We show that a ratio of Schur polynomials $s_λ/s_ρ$ associated to partitions $λ$ and $ρ$ such that $λ\subsetneqρ$ has a negative partial derivative at any point where all variables are positive. This is accomplished by establishing an injective map between sets of pairs of skew semistandard Young tableaux that preserves the product of the corresponding monomials. We use this result and the description of the first Laurent coefficient of the Hilbert series of the graded algebra of regular functions on a linear symplectic quotient by the circle to demonstrate that many such symplectic quotients are not graded regularly diffeomorphic. In addition, we give an upper bound for this Laurent coefficient in terms of the largest two weights of the circle representation and demonstrate that all but finitely many circle symplectic quotients of each dimension are not graded regularly diffeomorphic to linear symplectic quotients by $\operatorname{SU}_2$.

math.CO↗

arXiv2024

Factorization of spread polynomials

We present a proof of a conjecture of Goh and Wildberger on the factorization of the spread polynomials. We indicate how the factors can be effectively calculated and exhibit a connection to the factorization of Fibonacci numbers into primitive parts.

math.NT↗

arXiv2024

An exploration of connections and curvature in the presence of singularities

We develop the notions of connections and curvature for general Lie-Rinehart algebras without using smoothness assumptions on the base space. We present situations when a connection exists. E.g., this is the case when the underlying module is finitely generated. We show how the group of module automorphism acts as gauge transformations on the space of connections. When the underlying module is projective we define a version of the Chern character reproducing results of Hideki Ozeki. We discuss various examples of flat connections and the associated Maurer-Cartan equations. We provide examples of Levi-Civita connections on singular varieties and singular differential spaces with non-zero Riemannian curvature. The main observation is that for quotient singularities, even though the metric degenerates along strata, the poles of the Christoffel symbols are removable.

math.DG↗

arXiv2024

Hilbert measures on orbit spaces of coregular $\operatorname{O}_m$-modules

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 strata of the orbit space if and only if the number of copies of the defining representation of $\operatorname{O}_m$ is equal to $m$.

math.RT↗

arXiv2024

The symplectic form associated to a singular Poisson algebra

Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the singular case, however, derivations do not form a projective module and the nondegeneracy condition is more subtle. For a symplectic singularity one may naively ask if there is indeed an analogue of a symplectic form. We examine an example of a symplectic singularity, namely the double cone, and show that here such a symplectic form exists. We use the naive de Rham complex of a Lie-Rinehart algebra. Our analysis of the double cone uses Gröbner bases calculations. We also give an alternative construction of the symplectic form that generalizes to categorical quotients of cotangent lifted representations of finite groups. We use the same formulas to construct a symplectic form on the simple cone, seen as a Poisson differential space and generalize the construction to linear symplectic orbifolds. We present useful auxiliary results that enable to explicitly determine generators for the module of derivations an affine variety. The latter may be understood as a differential space.

math.AG↗

arXiv2024

Higher Koszul brackets on the cotangent complex

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbol k[x_1,x_2,\dots, x_n]/I$ where $\boldsymbol k$ is a field of characteristic $0$ and $I\subseteq \boldsymbol k[x_1,x_2,\dots, x_n]$ is an ideal. Assume that there is a Poisson bracket $\{\:,\:\}$ on $S$ such that $\{I,S\}\subseteq I$ and let us denote the induced bracket on $A$ by $\{\:,\:\}$ as well. It is well-known that $[\mathrm d x_i,\mathrm d x_j]:=\mathrm d\{x_i,x_j\}$ defines a Lie bracket on the $A$-module $Ω_{A|\boldsymbol k}$ of Kähler differentials making $(A,Ω_{A|\boldsymbol k})$ a Lie-Rinehart pair. Recall that $A$ is regular if and only if $Ω_{A|\boldsymbol k}$ is projective as an $A$-module. If $A$ is not regular, the cotangent complex $\mathbb L_{A|\boldsymbol k}$ may serve as a replacement for the $A$-module $Ω_{A|\boldsymbol k}$. We prove that there is a structure of an $L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$, compatible with the Lie-Rinehart pair $(A,Ω_{A|\boldsymbol k})$. The $L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$ actually comes from a $P_\infty$-algebra structure on the resolvent of the morphism $k[x_1,x_2,\dots, x_n]\to A$. We identify examples when this $L_\infty$-algebroid simplifies to a dg Lie algebroid. For aesthetic reasons we concentrate on cases when $ \boldsymbol k[x_1,x_2,\dots, x_n]$ carries a (possibly nonstandard) $\mathbb Z_{\ge 0}$-grading and both $I$ and $\{\:,\:\}$ are homogeneous.

math.AG↗

arXiv2024

A symmetric function approach to polynomial regression

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 $\sum _{i=1}^{n} a_{i} x^{d_{i}}$ for some integer exponents $d_{1} >d_{2} >\dotsc >d_{n} \geq 0$ and phrase the results using Schur polynomials. Even though the solution circumvents the well-known problems with the forward stability of the normal equation, it is only of practical value if $n$ is small because the number of terms in the formula grows rapidly with the number $m$ of data points. The formula can be evaluated essentially without rounding.

math.RA↗

arXiv2023

On the numerology of trigonometric polynomials

This note is about the observation that the various transition formulas between bases of trigonometric polynomials can be expressed in terms binomial coefficients. More specifically, we write the entries of the Chebyshev matrices $ T$ and $ U$ in terms of binomial coefficients. We remark that the inverses of certain submatrices of the Chebyshev matrix $ T$ correspond to classical power reduction formulas for trigonometric functions. Therefore the entries of $ T^{-1}$ are expressible in terms of binomial coefficients and powers of two. The inverses of certain submatrices of the Chebyshev matrix $ U$ appear in power reduction formulas involving the Catalan triangles. We could not spot them in the literature. As a corollary the Catalan triangles are interpreted as inverses of rather natural matrices of binomial coefficients. We explain the matrix inversions in terms of Riordan arrays. We solve the integral $ \int _{0}^{2π}\cos^{2m}( x)\sin^{2n}( x) dx$ and deduce a hypergeometric identity from the Fourier expansion of $\cos^{2m}(x)$ and $ \sin^{2n}( x)$. As a corollary we prove that super Catalan numbers are integers. We emphasize that from the point of view of these base changes between the trigonometric polynomials it is more natural to work with $2\cos(x)$ and $2\sin(x)$ instead of $\cos(x)$ and $\sin(x)$, since the transition matrices become invertible over the integers. We do a similar analysis for the spread polynomials. This enables us to make a conjecture of Goh and Wildberger more precise. In this exposition we give a mostly self-contained account of the matter, the prerequisites just being four semesters of calculus, linear algebra, elementary complex variables and the residue theorem.

math.HO↗

arXiv2023

Isomorphisms of Symplectic Torus Quotients

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 $μ\colon V\oplus V^*\to\mathfrak{g}^*$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N/\!\!/ G$. We show that $M$ determines $V\oplus V^*$ and $G$, up to isomorphism, if $\operatorname{codim}_N N_\mathrm{sing}\geq 4$. If $\operatorname{codim}_NN_\mathrm{sing}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G'\subset G$ and a faithful $1$-modular $G'$-submodule $V'\subset V$ with shell $N'$ such that $\operatorname{codim}_{N'}(N')_\mathrm{sing}\geq 4$. Moreover, there is a $G'$-equivariant morphism $N'\to N$ inducing an isomorphism $N'/\!\!/ G'\xrightarrow{\sim} N/\!\!/ G$. Thus, up to isomorphism, $M$ determines $V'\oplus (V')^*$ and $G'$, hence also $N'$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups.

math.SG↗

arXiv2023

Pivotal condensation and chemical balancing

I present a universal method, called pivotal condensation, for calculating stoichiometric factors of chemical reactions. It can be done by hand, even for rather complicated reactions. The main trick, which I call kernel pivotal condensation (ker pc), to calculate the kernel of a matrix might be of independent interest. The discussion is elaborated for matrices with entries in a principal ideal domain $R$. The ker pc calculates a basis with coefficients in $R$ for the kernel of a matrix, seen as e $Q$-vector space, where $Q$ is the quotient field of $R$. If $W$ is a free saturated $R$-submodule of $R^n$ I address the question how to modify an $R$-basis of the $Q$-vector subspace $Q\otimes _R W$ over the quotient field $Q$ to obtain a basis of the $R$-module $W$. I also indicate how one can solve inhomogeneous linear systems, invert matrices and determine the four subspaces using pivotal condensation. I formulate the balancing by inspection method that is widely used to reduce the size of a linear system arising in chemical balancing in mathematical language.

math.RA↗

arXiv2023

When does the zero fiber of the moment map have rational singularities?

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 find criteria for $N_V$ to have rational singularities and for the categorical quotient $N_V /\!\!/ G$ to have symplectic singularities, the latter results improving upon [HSS20]. It turns out that for ``most'' $G$-modules $V$, the shell $N_V$ has rational singularities. For the case of direct sums of classical representations of the classical groups, $N_V$ has rational singularities and $N_V /\!\!/ G$ has symplectic singularities if $N_V$ is a reduced and irreducible complete intersection. Another important special case is $V=p\,\mathfrak{g}$ (the direct sum of $p$ copies of the Lie algebra of $G$) where $p\geq 2$. We show that $N_V$ has rational singularities and that $N_V /\!\!/ G$ has symplectic singularities, improving upon results of [Bud19], [AA16], [Kap19] and [GH20]. Let $π=π_1(Σ)$ where $Σ$ is a closed Riemann surface of genus $p\geq 2$. Let $G$ be semisimple and let $\operatorname{Hom}(π,G)$ and $\mathscr X\!(π,G)$ be the corresponding representation variety and character variety. We show that $\operatorname{Hom}(π,G)$ is a complete intersection with rational singularities and that $\mathscr X\!(π,G)$ has symplectic singularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then the singularities of $\operatorname{Hom}(π,G)$ and $\mathscr X\!(π,G)$ are in codimension at least four and $\operatorname{Hom}(π,G)$ is locally factorial. If, in addition, $G$ is simply connected, then $\mathscr X\!(π,G)$ is locally factorial.

math.AG↗

arXiv2023

Higher Form Brackets for even Nambu-Poisson Algebras

Let $\boldsymbol{k}$ be a field of characteristic zero and $A=\boldsymbol{k}[x_{1},...,x_{n}]/I$ with $I=(f_{1},...,f_{k})$ be an affine algebra. We study Nambu-Poisson brackets on $A$ of arity $m\geq 2$, focusing on the case when $m$ is even. We construct an $L_{\infty}$-algebroid on the cotangent complex $\mathbb{L}_{A|\boldsymbol{k}}$, generalizing previous work on the case when $A$ is a Poisson algebra. This structure is referred to as the higher form brackets. The main tool is a $P_{\infty}$-structure on a resolvent $R$ of $A$. These $P_{\infty}$- and $L_{\infty}$-structures are merely $\mathbb Z_2$-graded for $m\neq 2$. We discuss several examples and propose a method to obtain new ones that we call the outer tensor product. We compare our higher form brackets with the form bracket of Vaisman. We introduce the notion of a Lie-Rinehart $m$-algebra, the form bracket of a Nambu-Poisson bracket of even arity being an example. We find a flat Nambu connection on the conormal module.

math.AG↗

arXiv2022

Gauge symmetry of linearised Nordström gravity and the dual spin two field theory

The field equations are proposed for the third rank tensor field with the hook Young diagram. The equations describe the irreducible spin two massless representation in any $d\geq 3$. The starting point of the construction is the linearised system of Einstein equations which includes the Nordström equation. This equation, being considered irrespectively to the rest of the Einstein system, corresponds to the topological field theory. The general solution is a pure gauge, modulo topological modes which we neglect in this article. We find the sequence of the reducible gauge transformations for the linearised Nordström equation, with the hook tensor being the initial gauge symmetry parameter. By substituting the general solution of the Nordström equation into the rest of the Einstein's system, we arrive at the field equations for the hook tensor. The degree of freedom number count confirms, it is the spin two theory.

hep-th↗

arXiv2022

Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups

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 decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series.

math.RA↗

arXiv2020

Hilbert series of symplectic quotients by the 2-torus

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the $2$-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at $t = 1$. We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples.

math.SG↗

arXiv2020

Constructing symplectomorphisms between symplectic torus quotients

We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic.

math.SG↗

arXiv2019

Symplectic quotients have symplectic singularities

Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V, G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or $\operatorname{SU}_2$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large.

math.SG↗

arXiv2019

Invariant Whitney Functions

A theorem of Gerald Schwarz [24, Thm. 1] says that for a linear action of a compact Lie group $G$ on a finite dimensional real vector space $V$ any smooth $G$-invariant function on $V$ can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set $Z\subset V$ fulfilling some regularity assumptions. In order to deal with the case when $Z$ is not $G$-stable we use the language of groupoids.

math.SG↗