On presentations of Brauer-type monoids

Ganna Kudryavtseva 1  and Volodymyr Mazorchuk 2
  • 1 Kyiv Taras Shevchenko University
  • 2 Uppsala University

Abstract

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.

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  • [1] J. Baez: “Link invariants of finite type and perturbation theory”, Lett. Math. Phys., Vol. 26(1), (1992), pp. 43–51. http://dx.doi.org/10.1007/BF00420517

  • [2] H. Barcelo and A. Ram: Combinatorial representation theory. New perspectives in algebraic combinatorics, Berkeley, CA, 1996–97, pp. 23–90.

  • [3] J. Birman: “New points of view in knot theory”, Bull. Amer. Math. Soc. (N.S.), Vol. 28(2), (1993), pp. 253–287.

  • [4] J. Birman and H. Wenzl: “Braids, link polynomials and a new algebra”, Trans. Amer. Math. Soc., Vol. 313(1), (1989), pp. 249–273. http://dx.doi.org/10.2307/2001074

  • [5] M. Bloss: “The partition algebra as a centralizer algebra of the alternating group”, Comm. Algebra, Vol. 33(7), (2005), pp. 2219–2229. http://dx.doi.org/10.1081/AGB-200063579

  • [6] R. Brauer: “On algebras which are connected with the semisimple continuous groups”, Ann. of Math. (2), Vol. 38(4), (1937), pp. 857–872. http://dx.doi.org/10.2307/1968843

  • [7] D. FitzGerald: “A presentation for the monoid of uniform block permutations”, Bull. Aus. Math. Soc., Vol. 68, (2003), pp. 317–324.

  • [8] D. FitzGerald and J. Leech: “Dual symmetric inverse monoids and representation theory”, J. Austral. Math. Soc. Ser. A, Vol. 64(3), (1998), pp. 345–367. http://dx.doi.org/10.1017/S1446788700039227

  • [9] T. Halverson and A. Ram: “Partition algebras”, European J. Comb., Vol. 26, (2005), pp. 869–921. http://dx.doi.org/10.1016/j.ejc.2004.06.005

  • [10] V.F.R. Jones: The Potts model and the symmetric group. Subfactors (Kyuzeso, 1993), World Sci. Publishing, River Edge, NJ, 1994, pp. 259–267.

  • [11] S. Kerov: “Realizations of representations of the Brauer semigroup”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Vol. 164, (1987); Differentsialnaya Geom. Gruppy Li i Mekh., Vol. IX, pp. 188–193, 199; translation in J. Soviet Math., Vol. 47(2), (1989), pp. 2503–2507.

  • [12] S. Lipscomb: Symmetric inverse semigroups. Mathematical Surveys and Monographs, Vol. 46, American Mathematical Society, Providence, RI, 1996.

  • [13] V. Maltcev: “Systems of generators, ideals and the principal series of the Brauer semigroup”, Proceedings of Kyiv University, Physical and Mathematical Sciences, Vol. 2, (2004), pp. 59–65.

  • [14] V. Maltcev: “On one inverse subsemigroups of the semigroup ℭn”, to appear in Proceedings of Kyiv University.

  • [15] V. Maltcev: On inverse partition semigroups IP x, preprint, Kyiv University, Kyiv, Ukraine, 2005.

  • [16] P. Martin: “Temperley-Lieb algebras for nonplanar statistical mechanics — the partition algebra construction”, J. Knot Theory Ramifications, Vol. 3(1), (1994), pp. 51–82. http://dx.doi.org/10.1142/S0218216594000071

  • [17] P. Martin: “The structure of the partition algebras”, J. Algebra, Vol. 183(2), (1996), pp. 319–358. http://dx.doi.org/10.1006/jabr.1996.0223

  • [18] P. Martin and A. Elgamal: “Ramified partition algebras”, Math. Z., Vol. 246(3), (2004), pp. 473–500. http://dx.doi.org/10.1007/s00209-003-0581-4

  • [19] P. Martin and D. Woodcock: “On central idempotents in the partition algebra”, J. Algebra, Vol. 217(1), (1999), pp. 156–169. http://dx.doi.org/10.1006/jabr.1998.7754

  • [20] V. Mazorchuk: “On the structure of Brauer semigroup and its partial analogue”, Problems in Algebra, Vol. 13, (1998), pp. 29–45.

  • [21] V. Mazorchuk: “Endomorphisms of B n, PB n, and ℭn”, Comm. Algebra, Vol. 30(7), (2002), pp. 3489–3513. http://dx.doi.org/10.1081/AGB-120004500

  • [22] M. Parvathi: “Signed partition algebras”, Comm. Algebra, Vol. 32(5), (2004), pp. 1865–1880. http://dx.doi.org/10.1081/AGB-120029909

  • [23] A. Vernitski: “A generalization of symmetric inverse semigroups”, preprint 2005.

  • [24] Ch. Xi: “Partition algebras are cellular”, Compositio Math., Vol. 119(1), (1999), pp. 99–109. http://dx.doi.org/10.1023/A:1001776125173

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