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SIGMA 14 (2018), 134, 18 pages arXiv:1804.09603
https://doi.org/10.3842/SIGMA.2018.134
A Product on Double Cosets of $B_\infty$
Pablo Gonzalez Pagotto
Institut Fourier, Université Grenoble Alpes, Grenoble, France
Received May 28, 2018, in final form December 14, 2018; Published online December 27, 2018
Abstract
For some infinite-dimensional groups $G$ and suitable subgroups $K$ there exists a monoid structure on the set $K\backslash G/K$ of double cosets of $G$ with respect to $K$. In this paper we show that the group $B_\infty$, of the braids with finitely many crossings on infinitely many strands, admits such a structure.
Key words:
Braid group; double cosets; Burau representation.
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References
-
Artin E., Braids and permutations, Ann. of Math. 48 (1947), 643-649.
-
Artin E., Theory of braids, Ann. of Math. 48 (1947), 101-126.
-
Birman J.S., Braids, links, and mapping class groups, Annals of Mathematics Studies, Vol. 82, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo, 1974.
-
Birman J.S., Brendle T.E., Braids: a survey, in Handbook of Knot Theory, Elsevier B.V., Amsterdam, 2005, 19-103, math.GT/0409205.
-
Dehornoy P., A fast method for comparing braids, Adv. Math. 125 (1997), 200-235.
-
Geck M., Pfeiffer G., Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs New Series, Vol. 21, The Clarendon Press, Oxford University Press, New York, 2000.
-
González-Meneses J., Geometric embeddings of braid groups do not merge conjugacy classes, Bol. Soc. Mat. Mex. 20 (2014), 297-305.
-
Neretin Yu.A., Categories of symmetries and infinite-dimensional groups, London Mathematical Society Monographs, New Series, Vol. 16, The Clarendon Press, Oxford University Press, New York, 1996.
-
Neretin Yu.A., On multiplication of double cosets for ${\rm GL}(\infty)$ over a finite field, arXiv:1310.1596.
-
Neretin Yu.A., Sphericity and multiplication of double cosets for infinite-dimensional classical groups, Funct. Anal. Appl. 45 (2011), 225-239, arXiv:1101.4759.
-
Neretin Yu.A., Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories, Int. Math. Res. Not. 2012 (2012), 501-523, arXiv:0909.4739.
-
Neretin Yu.A., Infinite symmetric groups and combinatorial constructions of topological field theory type, Russian Math. Surveys 70 (2015), 715-773, arXiv:1502.03472.
-
Neretin Yu.A., Several remarks on groups of automorphisms of free groups, J. Math. Sci. 215 (2016), 748-754, arXiv:1306.6035.
-
Neretin Yu.A., Combinatorial encodings of infinite symmetric groups and descriptions of semigroups of double cosets, J. Math. Sci. 232 (2018), 138-156, arXiv:1106.1161.
-
Okounkov A.Yu., Thoma's theorem and representations of the infinite bisymmetric group, Funct. Anal. Appl. 28 (1994), 100-107.
-
Ol'shankii G.I., Infinite-dimensional classical groups of finite $R$-rank: description of representations and asymptotic theory, Funct. Anal. Appl. 18 (1984), 22-34.
-
Ol'shankii G.I., Unitary representations of the group ${\rm SO}_{\rm o}(\infty, \infty)$ as limits of unitary representations of the groups ${\rm SO}_{\rm o}(n, \infty)$ as $n \rightarrow \infty$, Funct. Anal. Appl. 20 (1986), 292-301.
-
Ol'shankii G.I., Method of holomorphic extensions in the theory of unitary representations of infinite-dimensional classical groups, Funct. Anal. Appl. 22 (1988), 273-285.
-
Ol'shankii G.I., Unitary representations of $(G,K)$-pairs connected with the infinite symmetric group $S(\infty)$, Leningrad Math. J. 1 (1990), 983-1014.
-
Ol'shankii G.I., Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe, in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., Vol. 7, Gordon and Breach, New York, 1990, 269-463.
-
Ol'shankii G.I., On semigroups related to infinite-dimensional groups, in Topics in representation theory, Adv. Soviet Math., Vol. 2, Amer. Math. Soc., Providence, RI, 1991, 67-101.
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