By Helene Barcelo, Gil Kalai

This publication includes twenty-two papers awarded on the overseas convention in Combinatorics, held in Jerusalem in may possibly 1993. The papers describe a few of the newest advancements in algebraic combinatorics, enumeration, graph and hypergraph thought, combinatorial geometry, and geometry of polytopes and preparations. The papers are available to experts in addition to nonspecialists

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Extra resources for Jerusalem Combinatorics '93: An International Conference in Combinatorics, May 9-17, 1993, Jerusalem, Israel

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Proposition. Whenever we have three morphisms β : NX → NX , α : X1 → X , α : X → X1 , we have M α ◦ TrM N (β) ◦ α = TrN (N (α ) ◦ β ◦ N (α)) . In particular, the image if TrM N is a two-sided ideal in HomA ( , ). D2. 55. , M N (α) ◦ ηN,M (X1 ) = ηN,M (X) ◦ α . Similarly, we get εM,N (X1 ) ◦ M N (α ) = α ◦ εM,N (X ) . Using these equations, we obtain /X α X1 ηN,M (X1 ) ηN,M (X) M N X1 MNα / MNX εM,N (X1 ) εM,N (X ) Mβ /X1 O α XO / MNX MNα / MNX 1 α ◦ TrM N (β) ◦ α = α ◦ εM,N (X ) ◦ M (β) ◦ ηN,M (X) ◦ α = εM,N (X1 ) ◦ M (N (α ) ◦ β ◦ N (α)) ◦ ηN,M (X1 ) = TrM N (N (α ) ◦ β ◦ N (α)) .

Choose B := R, M :=A AR , N :=R AA and a, a := t(aa ). 1. The following conditions are equivalent. (i) A is strongly symmetric. (ii) R is A–split. 2. The following conditions are equivalent. (i) A is separable. (ii) A is R–split. Example : Induction–Restriction with a parabolic subalgebra. Let B be a parabolic subalgebra for A. Choose M :=A AB , N :=B AA , a, a := t(aa ) . Then B is always A–split, while A is B–split if and only if A is a summand of A ⊗B A in A ModA . If cA B = i ei ⊗B ei is the relative Casimir element, the “double relative trace” is  B   (A ⊗B A) → ZA TrA B: xj ⊗ yj → ei ( xj yj )ei   j i j Notice that the element 1 ⊗B 1 belongs to (A ⊗B A)B .

60. Lemma. A morphism X → X in A is M -split if and only if it factorizes through an M –split object of A. 61. Definition. Let A be an abelian category. The category Stab(A), is defined as follows: 1. the objects of Stab(A) are the objects of A, 2. , HomstA,M (X, X ) := HomA (X, X )/HomM A (X, X ) . Let A be an R-algebra. In the situation where A = A Mod, B = R Mod and A the biadjoint pair of functors is given by (IndA R , ResR ), we denote the corresponding stable category by A Stab. 36 SYMMETRIC ALGEBRAS Remarks.

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