## Two boundary Hecke Algebras and the combinatorics of type $C$

Last updated: 27 January 2015

## References

[ASu9710037] T. Arakawa and T. Suzuki, Duality between ${𝔰𝔩}_{n}\left(ℂ\right)$ and the degenerate affine Hecke algebra of type $A$, J. Algebra 209 (1998) 288–304. MR1652134 arXiv:q-alg/9710037

[Bon2772408] C. Bonnafé, Semicontinuity properties of Kazhdan-Lusztig cells, New Zealand J. of Math. 39 (2009) 171–192. MR2772408 arXiv:0808.3522

[Dau2012] Z. Daugherty, Degenerate two-boundary centralizer algebras, Pacific J. Math. 258 (2012), no. 1, 91-142. MR2972480 arXiv:1007.3950

[DRaPREP] Z. Daugherty and A. Ram, Representations of two-boundary Temperley-Lieb algebras, in preparation.

[Dri1990] V.G. Drinfel'd, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990) 321-342. MR1025154

[Eno2006] N. Enomoto, Classification of the irreducible representations of the affine Hecke algebra of type B2 with unequal parameters, J. Math. Kyoto Univ. 46 (2006) 259273. MR2284343 arXiv:math.RT/0505252

[Gec2004] M. Geck, Computing Kazhdan-Lusztig cells for unequal parameters, J. Algebra 281 (2004) 342–365, MR2091976.

[Gon2011] J. González-Meneses, Basic results on braid groups, Ann. Math. Blaise Pascal 18 (2011) 15–59. MR2830088, arXiv:1010.0321.

[dAl2009] J. de Gier and A. Nichols, The two-boundary Temperley-Lieb algebra, J. Algebra 321 (2009), no. 4, 1132–1167. MR2489894

[Gui2008] J. Guilhot, Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters, PhD Thesis, Université Claude Bernard-Lyon 1 and University of Aberdeen 2008, http://www.lmpt.univ-tours.fr/~guilhot/PhDf.pdf, arXiv:math/0702272.

[KLu1987] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987) 153–215. MR0862716

[KRa2002] C. Kriloff and A. Ram, Representations of graded Hecke algebras, Represent. Theory 6 (2002) 31–69. MR1915086

[Kat2009] S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009) 305-371. MR2524498 arXiv:math/0601155

[LRa1997] 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. in Math. 125 (1997), 1-94. MR1427801 (98c:20015)

[Lus1989] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635. MR 90e:16049

[Lus2003] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series 18, American Mathematical Society (2003). MR1974442

[Mac1354144] I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, London, 1988. MR1354144

[Mac2003] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157 Cambridge University Press, Cambridge (2003) x+175 pp. ISBN: 0-521-82472-9, MR1976581.

[Oka1998] S. Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra 205 (1998) 337–367. MR1632816

[OSo2010] E. Opdam and M. Solleveld, Discrete series characters for affine Hecke algebras and their formal degrees, Acta Math. 205 (2010) 105-187. MR2736154 arXiv:0804.0026

[ORa0401317] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$, in Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces Mumbai 2004, V.B. Mehta ed., Tata Institute of Fundamental Research, Narosa Publishing House, Amer. Math. Soc. (2007), 423–473. MR2348913 arXiv:math/0401317

[Ram2002] A. Ram, Representations of rank two affine Hecke algebras, in: Advances in Algebra and Geometry (University of Hyderabad Conference 2001), Hidustan Book Agency, New Delhi, India, 2002, pp. 57–91. MR1986143 arXiv:math/0401327

[Ram2003] A. Ram, Affine Hecke algebras and generalized standard Young tableaux, J. Algebra, 260 (2003), 367-415. MR1976700 arXiv:0401323

[Ree1997] M. Reeder, Nonstandard intertwining operators and the structure of unramified principal series representations, Forum. Math. 9 (1997) no. 4, 457–516. MR1457135

[RRa0401322] A. Ram and J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory, in A tribute to C.S. Seshadri: Perspectives in Geometry and Representation theory, V. Lakshimibai et al eds., Hindustan Book Agency, New Delhi (2003), 428-466, MR2017596, arXiv:math.RT/0401322.

[Sol2012] M. Solleveld, On the classification of irreducible representations of affine Hecke algebras with unequal parameters, Representation Theory 16 (2012) 1–87. MR2869018 arXiv:1008.0177

[Sta1986-2] R.P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 46 (1986) 103–113. MR0859302

[VVa1996] M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting, Comm. Math. Phys. 182 2 (1996) 469–483. MR1447301 arXiv:q-alg/9506026

## Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type $C$ Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).