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Topics in Computational Algebra

The most function of those lectures is first to in short survey the elemental con­ nection among the illustration idea of the symmetric team Sn and the idea of symmetric services and moment to teach how combinatorial tools that come up evidently within the concept of symmetric capabilities bring about effective algorithms to specific numerous prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

Additional resources for Algebra lineaire et tensorielle

Example text

Consider the two Q-lattices (α1 , z1 ) = (0, 2 −1) and (α2 , z2 ) = (0, −1). The composite ((α1 , z1 ), (α2 , z2 )) ◦ ((α2 , z2 ), (α1 , z1 )) is equal to the identity ((α1 , z1 ), (α1 , z1 )). We can also consider the composition √ √ ( −1 (α1 , z1 ), −1 (α2 , z2 )) ◦ ((α2 , z2 ), (α1 , z1 )), √ √ where −1 (α2 , z2 ) = (α2 , z2 ), but this is not the identity, since −1 (α1 , z1 ) = (α1 , z1 ). However, we can still deﬁne a convolution algebra for the quotient Z of (50), by restricting the convolution product of R2 to homogeneous functions of weight zero with C∗ -compact support, where a function f has weight k if it satisﬁes f (g, α, uλ) = λk f (g, α, u), ∀λ ∈ C∗ .

1 Tower Power If V is an algebraic variety – or a scheme or a stack – over a ﬁeld k, a “tower” T over V is a family Vi (i ∈ I) of ﬁnite (possibly branched) covers of V such that for any i, j ∈ I, there is a l ∈ I with Vl a cover of Vi and Vj . Thus, I is a partially ordered set. This gives a corresponding compatible system of covering maps Vi → V . In case of a tower over a pointed variety (V, v), one ﬁxes a point vi over v in each Vi . Even though Vi may not be irreducible, we shall allow ourselves to loosely refer to Vi as a variety.

The reﬁned theory of automorphisms of the modular ﬁeld does not enter in any way in the deﬁnition of the algebra. ˆ let Gα ⊂ GL+ (Q) be the set of For α ∈ M2 (Z), 2 ˆ Gα = {g ∈ GL+ 2 (Q) : gα ∈ M2 (Z)}. 40 Alain Connes, Matilde Marcolli, and Niranjan Ramachandran ˆ × H determines a Then, as shown in [10], an element y = (α, z) ∈ M2 (Z) unitary representation of the Hecke algebra A on the Hilbert space 2 (Γ \Gα ), f (gs−1 , sα, s(z)) ξ(s), ((πy f )ξ)(g) := ∀g ∈ Gα (63) s∈Γ \Gα for f ∈ A and ξ ∈ 2 (Γ \Gα ).