Nagoya Mathematical Journal

Commuting families in Hecke and Temperley-Lieb algebras

Tom Halverson, Manuela Mazzocco, and Arun Ram

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We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group $U_{h}\mathfrak{gl}_{n}$. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

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Nagoya Math. J., Volume 195 (2009), 125-152.

First available in Project Euclid: 14 September 2009

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Zentralblatt MATH identifier

Primary: 20G05: Representation theory
Secondary: 16G99: None of the above, but in this section 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs


Halverson, Tom; Mazzocco, Manuela; Ram, Arun. Commuting families in Hecke and Temperley-Lieb algebras. Nagoya Math. J. 195 (2009), 125--152.

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  • S. Ariki, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series 26, Amer. Math. Soc., Providence, RI, 2002. ISBN: 0-8218-3232-8, MR1911030 (2004b:17022)
  • I. Cherednik, A new interpretation of Gel'fand-Tzetlin bases, Duke Math. J., 54 (1987), no. 2, 563--577.
  • V. G. Drinfel'd, Almost cocommutative Hopf algebras, Leningrad Math. J., 1 (1990), 321--342.
  • J. Dixmier, Enveloping algebras, Graduate Studies in Mathematics 11, Amer. Math. Soc., Providence, RI, 1996. ISBN: 0-8218-0560-6, MR1393197 (97c:17010)
  • W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics 129, Springer-Verlag, New York 1991. ISBN: 0-387-97527-6; 0-387-97495-4, MR1153249 (93a:20069)
  • F. Goodman, P. de la Harpe, and V. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications 14, Springer-Verlag, New York, 1989.
  • J. Graham and G. Lehrer, Diagram algebras, Hecke algebras and decomposition numbers at roots of unity, Ann. Scient. Éc. Norm. Sup. $4^\mathrme$ série 36 (2003), no. 4, 479--524. MR2013924 (2004k:20007)
  • J. Graham and G. Lehrer, Cellular and diagram algebras in representation theory, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, 2004, pp. 141--173. MR2074593 (2005i:20005)
  • I. Grojnowski, Affine $\fsl_p$ controls the representation theory of the symmetric group and related Hecke algebras, arxiv:math.RT/9907129.
  • M. Jimbo, A $q$-analogue of $U(\fgl(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys., 11 (1986), no. 3, 247--252. MR0841713 (87k:17011)
  • A. Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics 163, Cambridge University Press, Cambridge, 2005. ISBN: 0-521-83703-0, MR2165457 (2007b:20022)
  • D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math., 87 (1987), no. 1, 153--215. MR0862716 (88d:11121)
  • R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras, Adv. Math., 125 (1997), 1--94. MR1427801 (98c:20015)
  • I. G. Macdonald, Symmetric functions and Hall polynomials, Second Edition, Oxford University Press, 1995.
  • R. Orellana and A. Ram, Affine braids, Markov traces and the category O, Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces, Mumbai 2004, Tata Institute of Fundamental Research (TIFR), Mumbai, Narosa Publishing House, 2007, pp. 423--473.
  • A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.), 2 (1996), no. 4, 581--605. MR1443185 (99g:20024)
  • A. Ram, Skew shape representations are irreducible, Combinatorial and Geometric representation theory (S.-J. Kang and K.-H. Lee, eds\.), Contemp. Math. 325, Amer. Math. Soc., 2003, pp. 161--189. MR1988991 (2004f:20014)
  • N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation, and invariants of links I, LOMI Preprint E-4-87 (1987).