The table of marks, the Kostka matrix, and the character table of the symmetric group

On the direct product of partial Burnside rings

“…The cardinality of the unit group of the PBR relative to P W of a finite Coxeter group W is 4 if W is a irreducible Coxeter group with type A, B, D, E 6 , E 7 , or E 8 (see [2], [5]).…”

“…Yoshida introduced a theory of generalized Burnside ring (GBR for short) with respect to a family of subgroups of G (see [10]). In 2015, the study of the unit group of the GBR relative to a collection of G was started by Idei and Oda (see [2]). Also it is discussed by [4] and [5].…”

Multiplicative induction and units for the ring of monomial representations

“…The unit group Ω(G) × of the Burnside ring Ω(G) is studied in many papers (see, e.g., [6,9,11,15,18,19,20,24,30,32]). Section 6 is devoted to a review of some well-known facts about Ω(G) × .…”

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

“…Recently, in [7], Idei and the first author have given a formula of a non-identity unit of Ω(S n , Y n ) which is described in terms of the Möbius function μ Y n on the poset (Y n , ≤) (see Eq. (2)).…”

“…This formula is presented in[7, Corollary 5.2]. We aim to show that α is included in the image by the tom Dieck homomorphism.…”