The decomposition numbers of the Hecke algebra of type 𝐸*₆

Geck, Meinolf

doi:10.1090/S0025-5718-1993-1195429-5

The decomposition numbers of the Hecke algebra of type $E^ \ast _ 6$
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Math. Comp. 61 (1993), 889-899 Request permission

Abstract:

Let ${E_6}(q)$ be the Chevalley group of type ${E_6}$, over a finite field with q elements, l be a prime not dividing q, and ${H_R}(q)$ be the endomorphism ring of the permutation representation (over a valuation ring R with residue class field of characteristic l) of ${E_6}(q)$ on the cosets of a standard Borel subgroup $B(q)$. Then the l-modular decomposition matrix ${D_l}$ of the algebra ${H_R}(q)$ is a submatrix of the l-modular decomposition matrix of the finite group ${E_6}(q)$. In this paper we determine the matrices ${D_l}$, for all l, q as above. For this purpose, we consider the generic Hecke algebra H associated with the finite Weyl group of type ${E_6}$ over the ring $A = \mathbb {Z}[v,{v^{ - 1}}]$ of Laurent polynomials in an indeterminate v, and calculate the decomposition matrices of H which are associated with specializations of v to roots of unity over $\mathbb {Q}$ or values in a finite field. The computations were done by using the computer algebra systems MAPLE and GAP.

References

Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307

Reference manual

Richard Dipper, Polynomial representations of finite general linear groups in nondescribing characteristic, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 343–370. MR 1112168
Meinolf Geck, On the decomposition numbers of the finite unitary groups in nondefining characteristic, Math. Z. 207 (1991), no. 1, 83–89. MR 1106814, DOI 10.1007/BF02571376
Meinolf Geck, On the classification of $l$-blocks of finite groups of Lie type, J. Algebra 151 (1992), no. 1, 180–191. MR 1182021, DOI 10.1016/0021-8693(92)90138-C
Meinolf Geck, Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), no. 10, 2937–2973. MR 1179271, DOI 10.1080/00927879208824499
Meinolf Geck and Klaus Lux, The decomposition numbers of the Hecke algebra of type $F_4$, Manuscripta Math. 70 (1991), no. 3, 285–306. MR 1089065, DOI 10.1007/BF02568379
Gerhard Hiss, Decomposition numbers of finite groups of Lie type in nondefining characteristic, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 405–418. MR 1112171, DOI 10.1007/978-3-0348-8658-1_{1}7
R. A. Parker, The computer calculation of modular characters (the meat-axe), Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 267–274. MR 760660

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