Finite Symplectic Groups Research Articles

Wavefront sets and descent method for finite symplectic groups

Wavefront sets and descent method for finite symplectic groups

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSLn(q) and PSp2n(q)

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSLn(q)(n≥4), PSL3(q)(q>2), or PSp2n(q)(n≥3), is isomorphic to a group algebra.

On parabolic subgroups of symplectic reflection groups

AbstractUsing Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

GALOIS AUTOMORPHISMS AND CLASSICAL GROUPS

In a previous work, the second-named author gave a complete description of the action of automorphisms on the ordinary irreducible characters of the finite symplectic groups. We generalise this in two directions. Firstly, using work of the first-named author, we give a complete description of the action of Galois automorphisms on irreducible characters. Secondly, we extend both descriptions to cover the case of special orthogonal groups. As a consequence, we obtain explicit descriptions for the character fields of symplectic and special orthogonal groups.

On the pronormality of subgroups of odd index in some direct products of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be pronormal in [Formula: see text] if [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text] for each [Formula: see text]. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product [Formula: see text] of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of [Formula: see text]. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text]. Thus, in this paper, we obtain a criterion of pronormality of a subgroup [Formula: see text] of odd index in a subgroup of [Formula: see text], where each [Formula: see text] is a prime and each [Formula: see text] acts naturally on [Formula: see text], such that [Formula: see text] projects onto [Formula: see text].

On the (2,3)-generation of the finite symplectic groups

This paper is a new important step towards the complete classification of the finite simple groups which are (2,3)-generated. In fact, we prove that the symplectic groups Sp2n(q) are (2,3)-generated for all n≥4. Because of the existing literature, this result implies that the groups PSp2n(q) are (2,3)-generated for all n≥2, with the exception of PSp4(2f) and PSp4(3f).

Unisingular representations of finite symplectic groups

Let G be a group and ρ an irreducible representation of G. Then ρ is called unisingular if has eigenvalue 1 for every In this paper we determine the unisingular 2-modular irreducible representations of the symplectic group over the field of two elements. A similar problem is solved for the maximal tori of G.

Equivariant Lagrangian Floer cohomology via semi-global Kuranishi structures

Using a simplified version of Kuranishi perturbation theory that we call semi-global Kuranishi structures, we give a definition of the equivariant Lagrangian Floer cohomology of a pair of Lagrangian submanifolds that are fixed under a finite symplectic group action and satisfy certain simplifying assumptions.

Hypergeometric sheaves and finite symplectic and unitary groups

Hypergeometric sheaves and finite symplectic and unitary groups

The 2-minimal subgroups of symplectic groups

For a finite group G , a subgroup P of G is 2-minimal if B < P , where B = N G ( S ) for some Sylow 2-subgroup S of G , and B is contained in a unique maximal subgroup of P . Here we give a detailed and explicit description of all the 2-minimal subgroups for finite symplectic groups defined over a field of odd characteristic.

THE INDUCTIVE BLOCKWISE ALPERIN WEIGHT CONDITION FOR TYPE AND THE PRIME

AbstractWe establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$ . As a main step, we derive a labelling set for the irreducible $2$ -Brauer characters of the finite symplectic groups $\operatorname {Sp}_{2n}(q)$ (with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.

Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for any $n \ge 2$, and others whose geometric monodromy groups are the finite general unitary groups $GU_n(q)$ for any odd $n \geq 3$.

Modular invariants of finite gluing groups

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow p-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic p-group acting on an elementary abelian p-group.

Classification of degenerations and Picard lattices of Kählerian K3 surfaces with symplectic automorphism group

In [1]–[6] we classified the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order. This classification was not considered for the remaining groups of small order (, , , , and ) because each of these cases requires very long and difficult considerations and calculations. Here we consider this classification for the dihedral group of order .

Classification of Degenerations and Picard Lattices of Kählerian K3 Surfaces with Symplectic Automorphism Group C4

In the author’s papers of 2013–2018, the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order were classified. For the remaining groups of small order—D6, C4, (C2)2, C3, C2, and C1—the classification was not completed because each of these cases requires very long and difficult considerations and calculations. The case of D6 was recently completely studied in the author’s paper of 2019. In the present paper an analogous complete classification is presented for the cyclic group C4 of order 4.

Local systems and finite unitary and symplectic groups

For powers q of any odd prime p and any integer n≥2, we exhibit explicit local systems, on the affine line A1 in characteristic p>0 if 2|n and on the affine plane A2 if 2∤n, whose geometric monodromy groups are the finite symplectic groups Sp2n(q). When n≥3 is odd, we show that the explicit rigid local systems on the affine line in characteristic p>0 constructed in [11] do have the special unitary groups SUn(q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups.

Rigid local systems and finite symplectic groups

For certain powers q of odd primes p, and certain integers n≥1, we exhibit explicit rigid local systems on the affine line in characteristic p>0 whose geometric and arithmetic monodromy groups are Sp(2n,q).

Two conjectures on the Weil representations of finite symplectic and unitary groups

We propose two conjectures on the tensor product of Weil representations with irreducible representations in finite symplectic and unitary groups.

Symmetric and alternating powers of Weil representations of finite symplectic groups

Symmetric and alternating powers of Weil representations of finite symplectic groups

Classification of Picard lattices of K3 surfaces

Using the results of our papers [1]–[4] on the classification of degenerations of Kählerian K3 surfaces, we classify the Picard lattices of Kählerian K3 surfaces. By classification we mean classification depending on their possible finite symplectic automorphism groups and their non- singular rational curves when the Picard lattice is negative definite.