Computing SL(2,C) Central Functions with Spin Networks

Let G=SL(2,C) and F_r be a rank r free group. Given an admissible weight in N^{3r-3}, there exists a class function defined on Hom(F_r,G) called a central function. We show that these functions admit a combinatorial description in terms of graphs called trace diagrams. We then describe two algorithms (implemented in Mathematica) to compute these functions.Comment: to appear in Geometriae Dedicat

Multivariate Poincar\'e series for algebras of SL2SL_2-invariants

Let \mathcal{C}_{\mathbi{d}}, \mathcal{I}_{\mathbi{d}}, \mathbi{d}{=}(d_1,d_2,..., d_n) be the algebras of join covariants and joint invariants of the $n$ binary forms of degrees $d_1,d_2,..., d_n.$ Formulas for computation of the multivariate Poincar\'e series \mathcal{P}(\mathcal{C}_{\mathbi{d}},z_1,z_2,...,z_n,t) and \mathcal{P}(\mathcal{I}_{\mathbi{d}},z_1,z_2,...,z_n) are found.Comment: 5 page

Subexponential estimations in Shirshov's height theorem (in English)

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that F_{2, m} is a 2-generated associative ring with the identity x^m=0. Is it true, that the nilpotency degree of F_{2, m} has exponential growth?" We show that the nilpotency degree of l-generated associative algebra with the identity x^d=0 is smaller than Psi(d,d,l), where Psi(n,d,l)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let l, n and d>n be positive integers. Then all the words over alphabet of cardinality l which length is greater than Psi(n,d,l) are either n-divided or contain d-th power of subword, where a word W is n-divided, if it can be represented in the following form W=W_0 W_1...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' means lexicographical order here. A. I. Shirshov proved that the set of non n-divided words over alphabet of cardinality l has bounded height h over the set Y consisting of all the words of degree <n. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A.G.Kolotov and in 1993 A.Ya.Belov obtained exponential estimation. We show, that h<Phi(n,l), where Phi(n,l) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.Comment: 21 pages, Russian version of the article is located at the link arXiv:1101.4909; Sbornik: Mathematics, 203:4 (2012), 534 -- 55

Фауната на паяциѢ (Araneae) въ България. V. Подразредъ Arachnomorphae, II клонъ Trionichia, семейство Agalenidae. Die Spinnenfauna Bulgariens V. Unterordnung Arachnomorphae, II. Gruppe Trionichia, Familie Agalenidae

Volume: 15Start Page: 33End Page: 6