This paper gives a Schur–Weyl duality approach to the representation theory of the affine Hecke algebras of type C with unequal parameters. The first step is to realize the affine braid group of type $C_k$ as the group of braids on $k$ strands with two poles. Generalizing familiar methods from the one pole (type A) case, this provides commuting actions of the quantum group $U_q\mathfrak{g}$ and the affine braid group of type $C_k$ on a tensor space $M\otimes N \otimes V^{\otimes k}$. Special cases provide Schur–Weyl pairings between the affine Hecke algebra of type $C_k$ and the quantum group of type $\mathfrak{gl}_n$, resulting in natural labelings of many representations of the affine Hecke algebras of type C by partitions. Following an analysis of the structure of weights of affine Hecke algebra representations (extending the one parameter case to the three parameter case necessary for affine Hecke algebras of type C), we provide an explicit identification of the affine Hecke algebra representations that appear in tensor space.
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Keywords: affine Hecke algebras, representations, Schur–Weyl duality
Daugherty, Zajj 1; Ram, Arun 2
CC-BY 4.0
@article{ART_2025__2_3_355_0,
author = {Daugherty, Zajj and Ram, Arun},
title = {Two boundary {Hecke} algebras and combinatorics of type $C$},
journal = {Annals of Representation Theory},
pages = {355--404},
publisher = {The Publishers of ART},
volume = {2},
number = {3},
year = {2025},
doi = {10.5802/art.27},
language = {en},
url = {https://art.centre-mersenne.org/articles/10.5802/art.27/}
}
TY - JOUR AU - Daugherty, Zajj AU - Ram, Arun TI - Two boundary Hecke algebras and combinatorics of type $C$ JO - Annals of Representation Theory PY - 2025 SP - 355 EP - 404 VL - 2 IS - 3 PB - The Publishers of ART UR - https://art.centre-mersenne.org/articles/10.5802/art.27/ DO - 10.5802/art.27 LA - en ID - ART_2025__2_3_355_0 ER -
%0 Journal Article %A Daugherty, Zajj %A Ram, Arun %T Two boundary Hecke algebras and combinatorics of type $C$ %J Annals of Representation Theory %D 2025 %P 355-404 %V 2 %N 3 %I The Publishers of ART %U https://art.centre-mersenne.org/articles/10.5802/art.27/ %R 10.5802/art.27 %G en %F ART_2025__2_3_355_0
Daugherty, Zajj; Ram, Arun. Two boundary Hecke algebras and combinatorics of type $C$. Annals of Representation Theory, Volume 2 (2025) no. 3, pp. 355-404. doi : 10.5802/art.27. https://art.centre-mersenne.org/articles/10.5802/art.27/
[1] Duality between and the degenerate affine Hecke algebra, J. Algebra, Volume 209 (1998) no. 1, pp. 288-304 | DOI | MR | Zbl
[2] Semicontinuity properties of Kazhdan–Lusztig cells, N. Z. J. Math., Volume 39 (2009), pp. 171-192 | MR | Zbl
[3] Algebraic Bethe ansatz for the open XXZ spin chain with non-diagonal boundary terms via symmetry, SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 19 (2023), Paper no. 046 | DOI | MR | Zbl
[4] Degenerate two-boundary centralizer algebras, Pac. J. Math., Volume 258 (2012) no. 1, pp. 91-142 | DOI | MR | Zbl
[5] Calibrated representations of two boundary Temperley–Lieb algebras (2020) | arXiv
[6] The two-boundary Temperley–Lieb algebra, J. Algebra, Volume 321 (2009) no. 4, pp. 1132-1167 | DOI | MR | Zbl
[7] Almost cocommutative Hopf algebras, Algebra Anal., Volume 1 (1989) no. 2, pp. 30-46 | MR | Zbl
[8] Classification of the irreducible representations of the affine Hecke algebra of type with unequal parameters, J. Math. Kyoto Univ., Volume 46 (2006) no. 2, pp. 259-273 | DOI | MR | Zbl
[9] Computing Kazhdan–Lusztig cells for unequal parameters, J. Algebra, Volume 281 (2004) no. 1, pp. 342-365 | DOI | MR | Zbl
[10] Basic results on braid groups, Ann. Math. Blaise Pascal, Volume 18 (2011) no. 1, pp. 15-59 | DOI | Numdam | MR | Zbl
[11] On the determination of Kazhdan–Lusztig cells for affine Weyl groups with unequal parameters, J. Algebra, Volume 318 (2007) no. 2, pp. 893-917 | DOI | MR | Zbl
[12] An exotic Deligne–Langlands correspondence for symplectic groups, Duke Math. J., Volume 148 (2009) no. 2, pp. 305-371 | DOI | MR | Zbl
[13] Proof of the Deligne–Langlands conjecture for Hecke algebras, Invent. Math., Volume 87 (1987) no. 1, pp. 153-215 | DOI | MR | Zbl
[14] Representations of graded Hecke algebras, Represent. Theory, Volume 6 (2002), pp. 31-69 | DOI | MR | Zbl
[15] A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman–Wenzl, and type A Iwahori–Hecke algebras, Adv. Math., Volume 125 (1997) no. 1, pp. 1-94 | DOI | MR | Zbl
[16] Affine Hecke algebras and their graded version, J. Am. Math. Soc., Volume 2 (1989) no. 3, pp. 599-635 | DOI | MR | Zbl
[17] Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, 1995 (with contributions by A. Zelevinsky, Oxford Science Publications) | DOI | MR | Zbl
[18] Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge University Press, 2003 | DOI | MR | Zbl
[19] Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra, Volume 205 (1998) no. 2, pp. 337-367 | DOI | MR | Zbl
[20] Discrete series characters for affine Hecke algebras and their formal degrees, Acta Math., Volume 205 (2010) no. 1, pp. 105-187 | DOI | MR | Zbl
[21] Affine braids, Markov traces and the category , Algebraic groups and homogeneous spaces (Tata Institute of Fundamental Research Studies in Mathematics), Volume 19, Tata Institute of Fundamental Research, 2007, pp. 423-473 | MR | Zbl
[22] Affine Hecke algebras and generalized standard Young tableaux, J. Algebra, Volume 260 (2003) no. 1, pp. 367-415 (Special issue celebrating the 80th birthday of Robert Steinberg) | DOI | MR | Zbl
[23] Representations of rank two affine Hecke algebras, Advances in algebra and geometry (Hyderabad, 2001) (Proceedings of the international conference on algebra and geometry), Hindustan Book Agency, 2003, pp. 57-91 | MR | Zbl
[24] Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory, A tribute to C. S. Seshadri (Chennai, 2002) (Trends in Mathematics), Birkhäuser, 2003, pp. 428-466 | MR | Zbl
[25] Nonstandard intertwining operators and the structure of unramified principal series representations, Forum Math., Volume 9 (1997) no. 4, pp. 457-516 | DOI | MR | Zbl
[26] On the classification of irreducible representations of affine Hecke algebras with unequal parameters, Represent. Theory, Volume 16 (2012), pp. 1-87 | DOI | MR | Zbl
[27] Symmetries of plane partitions, J. Comb. Theory, Ser. A, Volume 43 (1986) no. 1, pp. 103-113 | DOI | MR | Zbl
[28] Schur duality in the toroidal setting, Commun. Math. Phys., Volume 182 (1996) no. 2, pp. 469-483 | DOI | MR | Zbl
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