Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let R be a commutative ring and Φ be a finite (also called spherical) reduced irreducible root system of rank ≥2. I will denote by St(Φ,R) the Steinberg group of type Φ over R, i. e. the quotient of the free product ∐α∈ΦXα of root subgroups Xα=⟨xα(a)∣a∈R⟩ modulo Chevalley commutator formulae: [xα(a);xβ(b)]=∏iα+jβ∈Φxiα+jβ(Ni,jα,βaibj), i,j∈N; (here Ni,jα,β … Read more

Lusztig’s definition of quantum groups

In his book Introduction to quantum groups, Lusztig gives a definition (Def 3.1.1) of the rational form UQ(q)q that is rather different from the usual approach (see [1,Ch.9.1] for expample). As far as I understood, the translation goes as follows: Let g be a simple Lie algebra and I a set of simple roots in … Read more

Good range and fair range

Let G be a noncompact simple Lie group with complexified Lie algebra g. Fix a Cartan involution θ, which defines a maximal compact subgroup K of G. Take a θ-stable Cartan subalgebra h and a root system Δ. Denote by ρ half the sum of all positive roots. Suppose that q is a θ-stable parabolic … Read more

qq-Kostant partition function and flow polytopes?

The Kostant partition function is known to be related to volumes and Ehrhart polynomials of flow polytopes of graphs (see e.g. https://link.springer.com/article/10.1007/s00031-008-9019-8 or https://academic.oup.com/imrn/article-abstract/2015/3/830/649778). There is a natural q-analog of the Kostant partition function; for instance Lusztig’s q-analog of weight multiplicity can be defined in terms of this q-Kostant partition function (see e.g. https://arxiv.org/abs/1406.1453). (Note … Read more

About the geometry of the set of weights that is strongly linked to λ\lambda

Define η↑λ if η=λ or η=sα⋅λ<λ for some α∈Φ+. More generally, η↑λ if η=λ or η=sα1sα2⋯sαr⋅λ↑sα2⋯sαr⋅λ↑⋯↑sαrλ↑λ for some α1,α2,⋯,αr∈Φ+. Let λ∈h∗ and X(λ)={η∈h∗:η↑λ}. I would like to ask whether there is any geometry about X(λ) or the convex hull of X(λ). Answer AttributionSource : Link , Question Author : James Cheung , Answer Author : … Read more

Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?

Let Δ be a irreducible root system and Δ+ be its positive roots. We say a subset Δ′⊂Δ+ can generate the Weyl group if reflections of roots in Δ′ generate the whole Weyl group. If we simply consider a partition of Δ+ into disjoint union Δ+=Δ+1⨿Δ+2, then it is quite possible that neither Δ+1 nor … Read more

Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?

Let G be a complex Chevalley group (not necessarily adjoint type) with C–rank≥2 and let H be a normal subgroup of G(Z) with a finite index (so H is Zariski dense in G). Let T a maximal torus in G and B a Borel subgroup containing T, let wα1,…,wαn represent the simple reflections of the … Read more

Stability of infinite root systems with a long path in their Coxeter diagrams

Given a Cartan matrix associated to a Coxeter diagram, I can modify it by replacing one of the edges in the diagram with a long chain of vertices connected by simply laced edges; for example, this is where most of the infinite families of finite or affine diagrams come from. Given a positive root of … Read more

Eigenvalues and eigenvectors of the exceptional simple Lie group E6, E7, E8

What is the significance of the eigenvalues and eigenvectors of the exceptional simple Lie group root lattice to the Lie group or other mathematics branches? For example, E6, we have $$ \left( \begin{array}{cccccc} 2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & … Read more