arXiv subjects

Quinton Westrich

Publications and source records attributed to Quinton Westrich.

3 recordsLinked to original sources

arXiv2015

Discriminants and Higher K-energies on Polarized Kähler Manifolds

Given a compact polarized Kähler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of Kähler potentials in the Kähler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is defined on the Kähler class, correspond to metrics with harmonic $k$th Chern form. In this paper it is shown that the higher K-energy functionals, when restricted to the Bergman metrics, are expressible as the energies of certain pairs of vectors (tensors products of discriminants). Consequentially, we obtain results on the asymptotic behavior of these functionals along 1-parameter subgroups and their boundedness properties.

math.DG↗

arXiv2011

Young's Natural Representations of $\mathcal{S}_4$

We calculate all inequivalent irreducible representations of $§_4$ by specifying the matrices for adjacent transpositions and indicating how to obtain general permutations in $§_4$ from these transpositions. We employ standard Young tableaux methods as found in Sagan's \emph{The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions} (2001).

math.RT↗

arXiv2011

Polynomial Invariant Theory of the Classical Groups

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of basic invariants for the classical groups GL$(V)$, O$(n)$, and Sp$(n)$ for $n$ even. In the first half of the paper we set up relevant definitions and theorems for our search for the set of basic invariants, starting with linear algebraic groups and then discussing associative algebras. We then state and prove a monumental theorem that will allow us to proceed with hope: it says that the set of basic invariants is finite if $G$ is reductive. Finally we state without proof the First Fundamental Theorems, which aim to list explicitly the relevant sets of basic invariants, for the classical groups above. We end by commenting on some applications of invariant theory, on the history of its development, and stating a useful theorem in the appendix whose proof lies beyond the scope of this work.

math.GN↗