Isomorphisms of subcategories of fusion systems of blocks and Clifford theory

Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a {(G,N,c)}-Brauer pair {(R,f_{R})} consists of a p-subgroup R of G and a block {f_{R}} of {(kC_{N}(R))}. If Q is a defect group of c and {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))}, then {(Q,f_{Q})} is a {(G,N,c)}-Brauer pair. The {(G,N,c)}-Brauer pairs form a (finite) poset. Set {H=N_{G}(Q,f_{Q})} so that {(Q,f_{Q})} is an {(H,C_{N}(Q),f_{Q})}-Brauer pair. We extend Lemma 8.6.4 of the above book to show that if {(U,f_{U})} is a maximal {(G,N,c)}-Brauer pair containing {(Q,f_{Q})}, then {(U,f_{U})} is a maximal {(H,C_{N}(c),f_{Q})}-Brauer pair containing {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of {\mathcal{F}_{(U,f_{U})}(G,N,c)} and {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including {(Q,f_{Q})} and {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.

The Projective Character Tables of a Solvable Group 26:6×2

The Chevalley–Dickson simple group G24 of Lie type G2 over the Galois field GF4 and of order 251596800=212.33.52.7.13 has a class of maximal subgroups of the form 24+6:A5×3, where 24+6 is a special 2-group with center Z24+6=24. Since 24 is normal in 24+6:A5×3, the group 24+6:A5×3 can be constructed as a nonsplit extension group of the form G¯=24·26:A5×3. Two inertia factor groups, H1=26:A5×3 and H2=26:6×2, are obtained if G¯ acts on 24. In this paper, the author presents a method to compute all projective character tables of H2. These tables become very useful if one wants to construct the ordinary character table of G¯ by means of Fischer–Clifford theory. The method presented here is very effective to compute the irreducible projective character tables of a finite soluble group of manageable size.

Clifford theory of Weil representations of unitary groups

Let {\mathcal{O}} be an involutive discrete valuation ring with residue field of characteristic not 2. Let A be a quotient of {\mathcal{O}} by a nonzero power of its maximal ideal, and let {*} be the involution that A inherits from {\mathcal{O}} . We consider various unitary groups {\mathcal{U}_{m}(A)} of rank m over A, depending on the nature of {*} and the equivalence type of the underlying hermitian or skew hermitian form. Each group {\mathcal{U}_{m}(A)} gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil representation of {\mathcal{U}_{m}(A)} with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.

Fischer-Clifford Matrices and Character Table of the Maximal Subgroup (29:(L3(4)):2 of U6(2):2

The automorphism group U6(2):2 of the unitary group U6(2)≅Fi21 has a maximal subgroup G¯ of the form (29:(L3(4)):2 of order 20643840. In this paper, Fischer-Clifford theory is applied to the split extension group G¯ to construct its character table. Also, class fusion from G¯ into the parent group U6(2):2 is determined.

A VARIANT OF HARISH-CHANDRA FUNCTORS

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.