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Kashiwara, Masaki; Enomoto, Naoya

Symmetric crystals and affine Hecke algebras of type \(B\). (English) Zbl 1130.20008

Proc. Japan Acad., Ser. A 82, No. 8, 131-136 (2006).
Summary: The Lascoux-Leclerc-Thibon conjecture [A. Lascoux, B. Leclerc and J.-Y. Thibon, Commun. Math. Phys. 181, 205–263 (1996; Zbl 0874.17009)], reformulated and solved by S. Ariki [J. Math. Kyoto Univ. 36, No. 4, 789–808 (1996; Zbl 0888.20011)], asserts that the \(K\)-group of the representations of the affine Hecke algebras of type \(A\) is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra \(\mathfrak g\) where \(\mathfrak g\) is \(\mathfrak{gl}_\infty\) or the affine Lie algebra \(A^{(1)}_\ell\), and the irreducible representations correspond to the upper global bases. In this note, we formulate analogous conjectures for certain classes of irreducible representations of affine Hecke algebras of type \(B\).

Cited in 2 Reviews
Cited in 8 Documents

MSC:

20C08 Hecke algebras and their representations
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Keywords:

irreducible representations; crystal bases; affine Hecke algebras; LLT conjecture; decomposition matrices; Lascoux-Leclerc-Thibon conjecture; Lie algebras

Citations:

Zbl 0874.17009; Zbl 0888.20011
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\textit{M. Kashiwara} and \textit{N. Enomoto}, Proc. Japan Acad., Ser. A 82, No. 8, 131--136 (2006; Zbl 1130.20008)
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References:

[1] S. Ariki, On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\), J. Math. Kyoto Univ. 36 (1996), no. 4, 789-808. · Zbl 0888.20011
[2] I. Grojnowski, Affine \(\mathfrak{sl}_p\) controls the representation theory of the symmetric group and related Hecke algebras, arXive:math.RT/9907129. · Zbl 0879.17011
[3] A. Lascoux, Bernard Leclerc and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205-263. · Zbl 0874.17009
[4] M. Kashiwara, On crystal bases of the \(q\)-analogue of universal enveloping algebras, Duke Math. J. 63 (1991) 465-516. · Zbl 0739.17005
[5] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. · Zbl 0703.17008
[6] V. Miemietz, On representations of affine Hecke algebras of type B, Algebr. Represent. Theory. (to appear). · Zbl 1167.20005
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