By Hans Wilhelm
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BΣ as a right kC-module, where Σ is the top group, and where w ρ,0 I(D) is the augmentation ideal of kD. bΣ ρ,0 by the by the right regular action, and thus the quotient annihilator acts on kΣw ⊗ kBΣ ρ,0 , as an algebra. is actually isomorphic to kΣw ⊗ kBΣ ρ,0 We may prove a version of lemma 87 over k, using BrP : Lemma 89. We have an isomorphism of algebras, kE ∼ = kΣw ⊗ kBΣ . exye, for x ∈ kΣw , y ∈ kBΣ ρ,0 . Proof. bΣ ρ,0 = kBρ,w (∆P ). Likewise, kN f is a summand of kBΣ ρ,w as L-L-bimodules, where the complement has summands whose vertices do not contain ∆P .
QUASI-HEREDITARY SUBQUOTIENTS OF OBΣ ρ,w 53 Lemma 118. Suppose that Γ is an ideal of Ires . Then OBΣ ρ,w /IΓ is a quasihereditary algebra, with indexing poset Γ. The decomposition matrix of the algebra Σ OBΣ ρ,w /IΓ is the square submatrix of the decomposition matrix of OBρ,w , whose rows are indexed by elements of Γ. Proof. The generalised Schur algebra OBSρ,w /XΓ is a quasi-hereditary algebra, whose indexing poset is Γ. Over the ﬁeld k, the idempotent ξω sends to zero precisely those simple modules indexed by unrestricted partitions.
The following are open: When p = 2, what are the summands of kΣv eN ? What are their vertices and what are their sources ? When p = 2, what is the vertex of Σv kΣv eN ? What is the source ? CHAPTER 6 Ringel duality inside Rock blocks of symmetric groups We prove that a Rock block of a symmetric group, of arbitrary defect, possesses a family of internal symmetries, given as Ringel dualities between various subquotients (theorem 123). A. Green’s walk around the Brauer tree , we name it “a walk along the abacus”.