Tuesday October 27, 2020 11:00
Hông Vân Lê (Institute of Mathematics, Czech Academy of Sciences, Prague) The \(SL(9,\mathbb R)\)-orbit space of trivectors on \(\mathbb R^9\) and Galois cohomology Abstract: A classification of 3-vectors on \(\mathbb R^9\), understood as a description of the orbit space of the standard \(GL(9,\mathbb R)\)-action on \(\Lambda^3 \mathbb R^9\), is related to geometry defined by 3-forms in dimensions up to 9 and to the \(\theta\)-representation of the \(\mathbb Z_3\)-graded split Lie algebra \(e_{8(8)}\). In this talk I shall outline a method of classification of 3-vectors on \(\mathbb R^9\) based on Vinberg-Elashivili classification of 3-vectors on \(\mathbb C^9\) and on our joint work in progress with M. Borovoi and W. de Graaf, in which we utilize Galois cohomology to find real forms of a complex orbit. |
TBA
Martyna Maciaszczyk (Silesian University of Technology, Gliwice) Ideals and zero divisors in the ring \(T(\infty,F)\) Abstract: In this talk I give description of the ideals and zero divisors in the ring \(T(\infty,F)\) of infinite \(\mathbb N \times \mathbb N\) uppertriangular matrices over the field \(F\). I show solution of Suškevič's problem using strong linear independence. In the description of ideals I use the concept of zero pattern. Literature: 1. P. Vermes, Non-associative rings of infinite matrices, Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indag. Math. 14 (1952), 245-252. 2. A.K. Sushkevich, On an infinite algebra of triangular matrices, Kharkov. Gos. Univ. Uch. Zap. 34 = Zap. Mat. Otd. Fiz.-Mat. Fak. i Kharkov. Mat. Obshch. 22 (1950), 77-93 (in Russian). 3. W. Hołubowski, M. Maciaszczyk, S. Żurek, Note on Suškevič's problem on zero divisors, Comm. Algebra 45 (2017), no.8, 3274-3277. Bartłomiej Pawlik (Silesian University of Technology, Gliwice) Properties of Cayley graphs of Sylow subgroups of symmetric groups \(S_{2^n}\) on diagonal bases Abstract: I present the description of the polynomial (Kaloujnine) representation of Sylow subgroups of symmetric groups. Then, I discuss the selected results concerning Cayley graphs of above-mentioned groups in regards to the problem of the isomorphism of the graphs. Literature: 1. L. Kaloujnine, La structure des p-groupes de Sylow des groupes symétriques finis, Ann. Sci. Ecole Norm. Sup. 65 (1948), 239-276. 2. B. Pawlik, The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs, Algebra Discr. Math. 21 (2016), no.2, 264-281 3. B. Pawlik, The girth of Cayley graphs over Sylow 2-subgroups of the symmetric groups \(S_{2^n}\) with diagonal bases, J. Algebra Appl. 18 (2019), no.12, 1950237 4. A. Słupik, V. Sushchansky, Minimal generating sets and Cayley graphs of Sylow p-subgroups of finite symmetric groups, Algebra Discrete Math. 8 (2009), no.4, 167-184. |
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