Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


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.

pdf (479 kb)   tex (27 kb)

References

  1. Artin E., Braids and permutations, Ann. of Math. 48 (1947), 643-649.
  2. Artin E., Theory of braids, Ann. of Math. 48 (1947), 101-126.
  3. 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.
  4. Birman J.S., Brendle T.E., Braids: a survey, in Handbook of Knot Theory, Elsevier B.V., Amsterdam, 2005, 19-103, math.GT/0409205.
  5. Dehornoy P., A fast method for comparing braids, Adv. Math. 125 (1997), 200-235.
  6. 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.
  7. González-Meneses J., Geometric embeddings of braid groups do not merge conjugacy classes, Bol. Soc. Mat. Mex. 20 (2014), 297-305.
  8. 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.
  9. Neretin Yu.A., On multiplication of double cosets for ${\rm GL}(\infty)$ over a finite field, arXiv:1310.1596.
  10. Neretin Yu.A., Sphericity and multiplication of double cosets for infinite-dimensional classical groups, Funct. Anal. Appl. 45 (2011), 225-239, arXiv:1101.4759.
  11. 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.
  12. Neretin Yu.A., Infinite symmetric groups and combinatorial constructions of topological field theory type, Russian Math. Surveys 70 (2015), 715-773, arXiv:1502.03472.
  13. Neretin Yu.A., Several remarks on groups of automorphisms of free groups, J. Math. Sci. 215 (2016), 748-754, arXiv:1306.6035.
  14. 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.
  15. Okounkov A.Yu., Thoma's theorem and representations of the infinite bisymmetric group, Funct. Anal. Appl. 28 (1994), 100-107.
  16. 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.
  17. 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.
  18. 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.
  19. 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.
  20. 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.
  21. 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.

Previous article  Next article   Contents of Volume 14 (2018)