Narrow Results

In this paper, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors I such that

(1) for any e ≠ f ∈ I, the subVOA VOA(e, f) generated by e and f is isomorphic to either U_2B or U_3C; and

(2) the subgroup generated by the corresponding Miyamoto involutions {τ_e | e ∈ I} is isomorphic to the Weyl group of a root system of type A_n, D_n, E₆, E₇ or E₈.

The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of these vertex operator algebras are determined.

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊_n supported by 𝕊_n are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊_n supported by A to be finite dimensional.

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad(K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E₆/(SU(4) × SU(2) × U(1) × U(1)) and E₇/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2ℓ + 1)/(U(1) × U(p) × SO(2(ℓ - p - 1) + 1)) and SO(2ℓ)/(U(1) × U(p) × SO(2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

Given any quadruple $(N,P,Q,M)$ of ${II}_1$-factors with finite index, the notions of interior and exterior angles between $P$ and $Q$ were introduced in [An angle between intermediate subfactors and its rigidity, Trans. Amer. Math. Soc.371(8) (2019) 5973–5991]. We determine the possible values of these angles in terms of the cardinalities of the Weyl groups of the intermediate subfactors when $(N,P,Q,M)$ is an irreducible quadrilateral and the subfactors $N\subset P$ and $N\subset Q$ are both regular. For an arbitrary irreducible quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between $P$ and $Q$ are obtained when $N\subset P$ is regular, which enforce some restrictions on the index of $N\subset Q$ in terms of that of $N\subset P$.

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalize the Munn–Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn–Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type $A$ Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type $C$ and use them to obtain an explicit description of the character table for the type $C$ Renner monoid Hecke algebra.

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types B_n, C_n and D_n. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

We describe linear groups over an algebraically closed field in which the normalizer of a maximal torus splits over the torus. We describe linear groups over a finite field and their maximal tori in which the normalizer of the maximal torus splits over the torus.

For the exceptional Weyl group of type $F_4$, its Gröbner–Shirshov basis and the corresponding standard monomials are constructed.

We answer the question of splitting of the normalizer of a maximal torus in orthogonal groups of type $B_n$ and $D_n$ over finite fields and algebraically closed fields of positive characteristic.

Let $p$ be a fixed prime and ${𝔽}_q$ the finite field with $q=p^r$ for some $r\ge 1.$ In this paper, we investigate the absolute length function for certain reflection subgroups of the finite general linear group GL$({𝔽}_q^n)$ which are generalizations of Weyl groups of Cartan-type Lie algebras. An essential enumerative outcome is the computation of the number of elements with fixed length in the parabolic subgroup of GL$({𝔽}_q^n)$, where a generating function is employed.

Let (W,S) be a Weyl group of type B_n. In this paper, we find certain nontrivial distinguished involutions in the two-sided cell Ω_t of W with a-value for 1≤ 2t ≤ n and n even (resp., for 1≤ 2t < n and n odd). Besides, for 1≤ i₁< ⋯ < i_t≤ n-1, let L_{i1⋯ it} be a left cell of W with a-value t² and R(L_{i1⋯ it}) ={s_i1,…,s_it}. There are only two involutions y(i₁,i₁)⋯ y(i_t-1,i_t-1) s_it and y(i₁,i₁)⋯ y(i_t,i_t) in the left cell L_{i1⋯ it}. We prove that the former (resp., latter) is a distinguished involution in the left cell L_{i1⋯ it} for t odd (resp., even).

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A₁. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

We prove that every length on a simple group over a locally compact field, is either bounded or proper.

In this paper, we investigate the Fischer group $F_{i_{22}}$. This group is generated by a conjugacy class of involutions, any non-commuting pair of which has product of order 3. Such involutions are called transpositions and their conjugacy class is denoted by D. Subgroups generated by elements of D are called D-groups as they have been called by Enright [G. M. Enright, The structure and subgroups of the Fischer groups $F_{22}$ and $F_{23}$, Ph.D. thesis, University of Cambridge (1976)], Fischer embedded or 3-transposition groups. Here, we obtain the following main results:

• •The rank of the groups $F=3\cdot F_{i_{22}}$, the Weyl group W of type $E_6$ over fields of characteristic 2, the group M of shape $3^6⋊U_4(2)\cdot 2$ and the group $H_1=3{\mathrm{\Omega }}_7(3)$ are computed. If G is a 3-transposition group (Fischer embedded) for a class $D$ of transpositions, then $\text{rank}(G)$ is defined as $\text{max}\{|X\left|\right|X\subseteq D$, all elements in X commute$\}$.
• •We prove that the subgroups ${\mathrm{\Omega }}_8^{+}(2)\cdot S_3,3^5\cdot S_6,3\cdot U_4(3)\cdot 2$ and $S{U}_6(2)$ are Fischer embedded and we give an explicit construction for them. It is remarkable to mention that Enright studied certain D-groups under a very strong condition that is $O_2(G)\le Z(G)\ge O_3(G)$ and $G^{\prime }=G^{″}$ where G is the group generated by a proper subset E of D, $G^{\prime }$ is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.

The 2-partitions of the set of positive roots and of the root system of finite type or affine type are completely classified.