Two commuting operators associated with a tridiagonal pair

Let \K denote a field and let V denote a vector space over \K with finite positive dimension. We consider an ordered pair of linear transformations A:V\to V and A*:V \to V that satisfy the following four conditions: (i) Each of A,A* is diagonalizable; (ii) there exists an ordering {V_i}_{i=0}^d of the eigenspaces of A such that A*V_i\subseteq V_{i-1}+V_i+V_{i+1} for 0\leq i\leq d, where V_{-1}=0 and V_{d+1}=0; (iii) there exists an ordering {V*_i}_{i=0}^{\delta} of the eigenspaces of A* such that AV*_i\subseteq V*_{i-1}+V*_i+V*_{i+1} for 0\leq i\leq\delta, where V*_{-1}=0 and V*_{\delta+1}=0; (iv) there does not exist a subspace W of V such that AW\subseteq W, A*W\subseteq W, W\neq0, W\neq V. We call such a pair a TD pair on V. It is known that d=\delta; to avoid trivialities assume d\geq 1. We show that there exists a unique linear transformation \Delta:V\to V such that (\Delta -I)V*_i\subseteq V*_0+V*_1+...+V*_{i-1} and \Delta(V_i+V_{i+1}+...+V_d)=V_0 +V_{1}+...+V_{d-i} for 0\leq i \leq d. We show that there exists a unique linear transformation \Psi:V\to V such that \Psi V_i\subseteq V_{i-1}+V_i+V_{i+1} and (\Psi-\Lambda)V*_i\subseteq V*_0+V*_1+...+V*_{i-2} for 0\leq i\leq d, where \Lambda=(\Delta-I)(\theta_0-\theta_d)^{-1} and \theta_0 (resp \theta_d) denotes the eigenvalue of A associated with V_0 (resp V_d). We characterize \Delta,\Psi in several ways. There are two well-known decompositions of V called the first and second split decomposition. We discuss how \Delta,\Psi act on these decompositions. We also show how \Delta,\Psi relate to each other. Along this line we have two main results. Our first main result is that \Delta,\Psi commute. In the literature on TD pairs there is a scalar \beta used to describe the eigenvalues. Our second main result is that each of \Delta^{\pm 1} is a polynomial of degree d in \Psi, under a minor assumption on \beta.Comment: 36 page

Arithmetic aspects of the Burkhardt quartic threefold

We show that the Burkhardt quartic threefold is rational over any field of characteristic distinct from 3. We compute its zeta function over finite fields. We realize one of its moduli interpretations explicitly by determining a model for the universal genus 2 curve over it, as a double cover of the projective line. We show that the j-planes in the Burkhardt quartic mark the order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse pencil on a j-plane gives rise to the universal curve as a discriminant of a cubic genus one cover.Comment: 22 pages. Amended references to more properly credit the classical literature (Coble in particular

Filtered screens and augmented Teichm\"uller space

We study a new bordification of the decorated Teichm\"uller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield short curves having vanishing length in the underlying surface F. As a result, an appropriate quotient of this space of filtered screens on F yields a decorated augmented Teichm\"uller space which is shown to admit a CW decomposition that naturally projects to the augmented Teichm\"uller space by forgetting decorations and whose strata are indexed by a new object termed partially oriented stratum graphs.Comment: Final version to appear in Geometriae Dedicat

{\Gamma}-species, quotients, and graph enumeration

The theory of {\Gamma}-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees.Comment: 84 pages, 10 figures, dissertatio