The staircase representation of S₆ and the regular representation of its 2-Sylow subgroup

We study the irreducible representation V3,2,1 of the symmetric group S6 associated with the staircase partition (3, 2, 1). We prove that its restriction to a Sylow 2-subgroup P = Syl2(S6) ∼= D8 ×Z2 is isomorphic to the regular representation CP (Regularity Theorem). This isomorphism is a consequence of a combinatorial identity: dim V3,2,1 = 16 = |P|, whichin turn follows from the fact that all hook lengths of the self-conjugate staircase partition areodd. More generally, we prove a p-Core Vanishing Theorem: for any prime p and any p-corepartition λ ⊢ n, the character χλ vanishes on every permutation whose cycle type contains apart divisible by p. As a consequence, the restriction of Vλ to a Sylow p-subgroup is alwaysa multiple of the regular representation. The staircase partitions (2-cores) are a special case;among them, k = 3 (i.e. S6) is the unique non-trivial instance with V |Syl2∼= CSyl2.From the Regularity Theorem we derive a canonical decomposition CP = Vab ⊕ Vnab withdim Vab = dim Vnab = 8, where Vab is the direct sum of the eight linear characters of P(corresponding to the abelianization P/P, P ∼= (Z2)3), and Vnab carries the two faithful 2-dimensional irreducible representations (originating from the dihedral factor D8). The projector πab =12(Id +La,d), where a, d is the unique non-trivial element of P, P, realizes this decomposition explicitly.We also determine the complete action of Out(S6) ∼= Z2 on the irreducible representations:ve are xed (dimensions 1, 1, 9, 9, 16) and three pairs are swapped. The two extensions ofV3,2,1 to Aut(S6) ∼= S6.2 have trace Tr(Mφ) = ±1 on the outer element, correcting a claim inthe literature. All results are verified by independent computation in GAP; the verification suite is publiclyavailable.