When the annihilator graph of a commutative ring is planar or toroidal?

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R)* = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if  ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified

Root and weight semigroup rings for signed posets

We consider a pair of semigroups associated to a signed poset, called the root semigroup and the weight semigroup, and their semigroup rings, $R_P^\mathrm{rt}$ and $R_P^\mathrm{wt}$, respectively. Theorem 4.1.5 gives generators for the toric ideal of affine semigroup rings associated to signed posets and, more generally, oriented signed graphs. These are the subrings of Laurent polynomials generated by monomials of the form $t_i^{\pm 1},t_i^{\pm 2},t_i^{\pm 1}t_j^{\pm 1}$. This result appears to be new and generalizes work of Boussicault, F\'eray, Lascoux and Reiner, of Gitler, Reyes, and Villarreal, and of Villarreal. Theorem 4.2.12 shows that strongly planar signed posets $P$ have rings $R_P^\mathrm{rt}$, $R_{P^{\scriptscriptstyle\vee}}$ which are complete intersections, with Corollary 4.2.20 showing how to compute $\Psi_P$ in this case. Theorem 5.2.3 gives a Gr\"obner basis for the toric ideal of $R_P^{\mathrm{wt}}$ in type B, generalizing Proposition 6.4 of F\'eray and Reiner. Theorems 5.3.10 and 5.3.1 give two characterizations (via forbidden subposets versus via inductive constructions) of the situation where this Gr\"obner basis gives a complete intersection presentation for its initial ideal, generalizing Theorems 10.5 and 10.6 of F\'eray and Reiner.Comment: 170 pages; 63 figures; PhD Dissertation, University of Minnesota, August 201

Geometric complexity theory, tensor rank, and Littlewood-Richardson coefficients

Diese Arbeit führt gründlich in die Geometrische Komplexitätstheorie ein, ein Ansatz, um untere Berechnungskomplexitätsschranken mittels Methoden aus der algebraischen Geometrie und Darstellungstheorie zu finden. Danach konzentrieren wir uns auf die relevanten darstellungstheoretischen Multiplizitäten, und zwar auf Plethysmenkoeffizienten, Kronecker-Koeffizienten und Littlewood-Richardson-Koeffizienten. Diese Multiplizitäten haben eine Beschreibung als Dimensionen von Höchstgewichtsvektorräumen, für welche konkrete Basen nur im Littlewood-Richardson-Fall bekannt sind.Durch explizite Konstruktion von Höchstgewichtsvektoren können wir zeigen, dass der Grenzrang der m x m Matrixmultiplikation mindestens 3 m^2 - 2 ist, und der Grenzrang der 2 x 2 Matrixmultiplikation genau sieben ist. Dies liefert einen neuen Beweis für ein Ergebnis von Landsberg (J. Amer. Math. Soc., 19:447-459, 2005).Desweiteren erhalten wir Nichtverschwindungsresultate für rechteckige Kronecker-Koeffizienten und wir beweisen eine Vermutung von Weintraub (J. Algebra, 129 (1): 103-114, 1990) uber das Nicht-Verschwinden von Plethysmen-koeffizienten von geraden Partitionen.Unsere eingehenden Untersuchungen zu Littlewood-Richardson-Koeffizienten c_We provide a thorough introduction to Geometric Complexity Theory, an approach towards computational complexity lower bounds via methods from algebraic geometry and representation theory. Then we focus on the relevant representation theoretic multiplicities, namely plethysm coefficients, Kronecker coefficients, and Littlewood-Richardson coefficients. These multiplicities can be described as dimensions of highest weight vector spaces for which explicit bases are known only in the Littlewood-Richardson case.By explicit construction of highest weight vectors we can show that the border rank of m x m matrix multiplication is a least 3 m^2 - 2 and the border rank of 2 x 2 matrix multiplication is exactly seven. The latter gives a new proof of a result by Landsberg (J. Amer. Math. Soc., 19:447-459, 2005).Moreover, we obtain new nonvanishing results for rectangular Kronecker coefficients and we prove a conjecture by Weintraub (J. Algebra, 129 (1): 103-114, 1990) about the nonvanishing of plethysm coefficients of even partitions.Our in-depth study of Littlewood-Richardson coefficients c_Tag der Verteidigung: 18.10.2012Paderborn, Univ., Diss., 201