The units of a partial Burnside ring relative to the Young subgroups of a symmetric group

The unit group of a partial Burnside ring relative to the Young subgroups of the symmetric group SnSn on n letters is included in the image by the tom Dieck homomorphism. As a consequence of this fact, the alternating character νnνn of SnSn is expressed explicitly as a ZZ-linear combinations of permutation characters associated with finite left SnSn-sets Sn/YSn/Y for the Young subgroups Y.Mathematics Subject Classification. Primary 19A22, Secondary 20B35; 20C15; 20C3

On a homotopy relation between the 2-local geometry and the Bouc complex for the sporadic group Co3

We study the homotopy relation between the standard 2-local geometry and the Bouc complex for the sporadic group Co3. We also give a result concerning the relative projectivity of the reduced Lefschetz module associated to the aformentioned 2-local geometry.Comment: 20 page

Euler characteristics of p-subgroup categories

We compute Euler characteristics of p-subgroup categories of finite groupsComment: 21 pages. Updated and expanded version of earlier submissio

Locally Determined Functions and Alperin's Conjecture

For a fixed prime number p, Alperin’s conjecture asserts essentially that the function k(G) − z(G) should be p-locally determined, where k(G) is the number of absolutely irreducible representations of a finite group G in characteristic zero, and z(G) is the number of those representations whose dimension is a multiple of |G|p, the p-part of the order of the group. Equivalently, the conjecture also says that the function `(G) − z(G) should be p-locally determined, where `(G) is this time the number of absolutely irreducible representations of G in characteristic p, and z(G) is the number of those representations which are both simple and projective modules over the group algebra, in other words the number of blocks of G of defect zero (this number z(G) is well-known to be equal to the one defined above). We first consider the question of a precise mathematical definition of p-local determination. We give a combinatorial definition involving all subgroups of G and procede to show that any function f on the poset of subgroups of G, which is constant on conjugacy classes of subgroups, decomposes uniquely as the sum of two functions f = fp+fp ′ where fp is p-locally determined and fp ′ vanishes on p-local subgroup

Lefschetz invariants and Young characters for representations of the hyperoctahedral groups

The ring R(Bn) of virtual C-characters of the hyperoctahedral group Bnhas two Z-bases consisting of permutation characters, and the ring structureassociated with each basis of them defines a partial Burnside ring of whichR(Bn) is a homomorphic image. In particular, the concept of Young charactersof Bn arises from a certain set Un of subgroups of Bn, and the Z-basis of R(Bn)consisting of Young characters, which is presented by L. Geissinger and D.Kinch [7], forces R(Bn) to be isomorphic to a partial Burnside ring Ω(Bn; Un).The linear C-characters of Bn are analyzed with reduced Lefschetz invariantswhich characterize the unit group of Ω(Bn; Un). The parabolic Burnside ringPB(Bn) is a subring of Ω(Bn; Un), and the unit group of PB(Bn) is isomorphicto the four group. The unit group of the parabolic Burnside ring of the even-signed permutation group Dn is also isomorphic to the four group

Multiplicative induction and units for the ring of monomial representations

Let G be a finite group, and let A be a finite abelian G-group. For each subgroup H of G, Ω(H;A) denotes the ring of monomial representations of H with coefficients in A, which is a generalization of the Burnside ring Ω(H) of H. We research the multiplicative induction map Ω(H;A) → Ω(G;A) derived from the tensor induction map Ω(H) → Ω(G), and also research the unit group of Ω(G;A). The results are explained in terms of the first cohomology groups H1(K;A) for K ≤ G. We see that tensor induction for 1-cocycles plays a crucial role in a description of multiplicative induction. The unit group of Ω(G;A) is identified as a finitely generated abelian group. We especially study the group of torsion units of Ω(G;A), and study the unit group of Ω(G) as well