Exceptional Weyl Research Articles

Brane transitions from exceptional groups

It is a well-known result by Hanany and Witten that, when two five-branes move across each other, D3-branes stretching between them are generated. Later the same brane configurations played a crucial role in understanding the worldvolume theory of multiple M2-branes. Recently the partition function of multiple M2-branes was transformed to the Fredholm determinant for quantum algebraic curves, where the characteristic 3/2 power law of degrees of freedom is reproduced and the determinant enjoys large symmetries given by exceptional Weyl groups. The large exceptional Weyl group reproduces the Hanany-Witten brane transitions and, besides, contains brane transitions unknown previously. Aiming at understanding the new brane transitions better, we generalize our previous study on the D5 quantum curve to the E7 case, which requires delicate handling of degeneracies. By combining the results of these two cases, we propose a “local” rule for the brane transitions.

The Tutte polynomial of symmetric hyperplane arrangements

The Tutte polynomial is originally a bivariate polynomial which enumerates the colorings of a graph and of its dual graph. Ardila extended in 2007 the definition of the Tutte polynomial on the real hyperplane arrangements. He particularly computed the Tutte polynomials of the hyperplane arrangements associated to the classical Weyl groups. Those associated to the exceptional Weyl groups were computed by De Concini and Procesi one year later. This paper has two objectives: On the one side, we extend the Tutte polynomial computing to the complex hyperplane arrangements. On the other side, we introduce a wider class of hyperplane arrangements which is that of the symmetric hyperplane arrangements. Computing the Tutte polynomial of a symmetric hyperplane arrangement permits us to deduce the Tutte polynomials of some hyperplane arrangements, particularly of those associated to the imprimitive reflection groups.

Standard monomials for the Weyl group F4

For the exceptional Weyl group of type [Formula: see text], its Gröbner–Shirshov basis and the corresponding standard monomials are constructed.

Standard monomials for the Weyl group F 4

The Weyl group of a root system coming from the Lie theory is defined to be the group generated by reflections with respect to the roots. We have three series of finite Weyl groups and five exceptional Weyl groups. For the Weyl groups of classical and some exceptional types, their Gröbner-Shirshov bases were constructed in [3, 7, 8].

Coinvariant algebras and fake degrees for spin Weyl groups of exceptional type

We compute all the spin fake degrees for the exceptional Weyl groups, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in its spin coinvariant algebra. The spin fake degrees are all shown to be palindromic polynomials.

Coinvariant algebras and fake degrees for spin Weyl groups of classical type

AbstractThe coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed by Steinberg, Lusztig and Beynon–Lusztig. In this paper we formulate a notion of spin coinvariant algebra for every Weyl group. Then we compute all the spin fake degrees for each classical Weyl group, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in the spin coinvariant algebra. The spin fake degrees for the exceptional Weyl groups are given in a sequel.

Elliptic hypergeometric functions and Calogero-Sutherland-type models

We consider an elliptic analogue of the Gauss hypergeometric function and two of its multivariate generalizations. We describe their relation to elliptic beta integrals, the exceptional Weyl group E7, the elliptic hypergeometric equation, and Calogero-Sutherland-type models.

Character tables of commutative hecke algebras associated with exceptional weyl groups

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups when they are commutative. In the case when Weyl groups are of classical type, they are already known in [D.1] and [D.2]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with exceptional Weyl groups.

Character tables of commutative hecke algebras associated with finite chevalley groups of exceptional type

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness

Decomposition numbers for spin characters of exceptional Weyl groups of type $E_n$

Decomposition numbers for spin characters of exceptional Weyl groups of type $E_n$

Combinatorics and poincaré polynomials of hyperplane complements for exceptional weyl groups

Combinatorics and poincaré polynomials of hyperplane complements for exceptional weyl groups

Projective characters of the rotation subgroup of the weyl group of typeF4

In an earlier paper [l0], the first author determined the projective characters of rotation subgroups W+(Φ) of the exceptional Weyl groups W(Φ), Φ=El (l=6,7,8). For the case Φ=F4, only those characters of W+(F4) whose factor set α corresponds to the basic spin representations determined in [9] were obtained. In the present paper we obtain the irreducible characters of W+(F4)corresponding to the remaining classes of factor sets, thereby completing the work in [lo]. We also give explicit constructions for some of the projective representations.

On the reflection representation in Springer's theory

The theory of arrangements of hyperplanes allows to attach to every parabolic subgroup of a finite Coxeter group some numbers which look very much like exponents of the Coxeter group. In the case of Weyl groups similar numbers arise from the character theory of finite groups of Lie type, and more generally from the theory of Springer's representations. For exceptional Weyl groups these numbers were known to coincide, at least if the characteristic is not a small prime. Lehrer and Shoji [8] have shown that in characteristic 0 the same is true for classical Weyl groups, by computing the multiplicity of the reflection representation in the Springer representations associated to various nilpotent orbits. According to a note added in proof, they can handle all nilpotent orbits which are relevant to the connection with arrangements of hyperplanes, but some nilpotent orbits still evade their investigation.

Projective characters of rotation subgroups of exceptional weyl groups

Projective characters of rotation subgroups of exceptional weyl groups

Decomposition numbers of exceptional Weyl groups, I

The decomposition numbers of the exceptional Weyl groups for the prime number p = 3 are obtained; other relevant prime numbers will be considered in forthcoming publications. General references on Weyl groups, the characters of the individual groups and on the modular representation theory required are given. The technique used for calculating the decomposition matrices are those described by James and Kerber together with the results of Brauer. Notation is introduced and then each individual group is considered and the results tabulated. For more details one may consult Khosraviyani, Ph.D. thesis, University College of Wales, May 1981.

Some numerical results on the characters of exceptional Weyl groups

1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading . LetI be the ideal in S generated by the w-invariant elements in . The quotient algebra inherits the W-action and also a grading is the image of Sk under S → S̅. It is well known that the W-module S̅ is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, S̅k = 0 for large k. (More precisely, S̅k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module S̅k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in qcan sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.

Projective characters of exceptional Weyl groups

Projective characters of exceptional Weyl groups

On the Schur Indices of Characters of the Exceptional Weyl Groups

We will be concerned in this paper with the irreducible characters of the Weyl groups of type E6, E7, and E8. Character tables for these groups have been given by J. S. Frame [6, 7]. In all cases, the characters are rationalvalued. It is fairly easy to show that each character has a real splitting field. The object of this paper is to show that each character has a rational representation. Other authors have shown that the same results hold for other irreducible Weyl groups. A. Young [13] studied the families of groups of type A, and B, as permutation groups. He devised a method for constructing all representations and showed that they could be constructed over the rational field Q. W. Specht [11] did the same for the family of groups of type D. This result is easily shown for the group of type G2, which is dihedral of order 12. T. Kondo [9] presented a character table for the group of type F4 and showed that each character has a rational representation. Our present results complete the proof that any representation of a Weyl group of a finite dimensional semi-simple complex Lie algebra is equivalent to a rational representation. Our methods are character-theory methods and we restate our results in terms of the Schur index. Let X be an irreducible character of a group G and let F be a field. The Schur index m,(X) of X over F is the smallest positive integer m such that mX is a character afforded by an F(X)-representation of G.