Reflection groups and 3d $$ \mathcal{N} $$> 6 SCFTs

AbstractRecent work has shown that every 3D root system allows the construction of a corresponding 4D root system via an ‘induction theorem’. In this paper, we look at the icosahedral case of $$H_3\rightarrow H_4$$ H 3 → H 4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan–Dieudonné theorem, giving a simple construction of the $${\mathrm {Pin}}$$ Pin and $${\mathrm {Spin}}$$ Spin covers. Using this connection with $$H_3$$ H 3 via the induction theorem sheds light on geometric aspects of the $$H_4$$ H 4 root system (the 600-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of $${\mathrm {Cl}}(3)$$ Cl ( 3 ) , including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.

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The Tutte polynomial of symmetric hyperplane arrangements

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216520500042 ◽

2020 ◽

Vol 29 (03) ◽

pp. 2050004

Author(s):

Hery Randriamaro

Keyword(s):

Hyperplane Arrangement ◽

Dual Graph ◽

Tutte Polynomial ◽

Hyperplane Arrangements ◽

Weyl Groups ◽

Reflection Groups ◽

Tutte Polynomials ◽

One Year ◽

The One ◽

Definition Of

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.

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Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

Nagoya Mathematical Journal ◽

10.1017/s0027763000009880 ◽

2010 ◽

Vol 197 ◽

pp. 175-212

Author(s):

Maria Chlouveraki

Keyword(s):

Infinite Series ◽

Hecke Algebras ◽

Weyl Groups ◽

Reflection Groups ◽

Complex Reflection ◽

Complex Reflection Groups ◽

Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

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