Jürgen Müller's Algebraic combinatorics PDF

Jürgen Müller's Algebraic combinatorics PDF

By Jürgen Müller

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Extra info for Algebraic combinatorics

Example text

A) Let X be a locally finite partially ordered set, and let ζ ∈ A(X) be the zeta function of X, that is the indicator function of the partial order, given by ζ(x, y) = 1 whenever x ≤ y ∈ X, and ζ(x, y) = 0 whenever x ≤ y ∈ X. Note that restricting ζ to an interval in X just yields the zeta function of the interval. The zeta function is related to chains: k i) Induction on k ∈ N yields ζ k (x, y) = x=z0 ≤z1 ≤···≤zk =y i=1 ζ(zi−1 , zi ) = k x=z0 ≤z1 ≤···≤zk =y 1, for x ≤ y ∈ X. Thus for k ∈ N0 we infer that ζ (x, y) ∈ N0 is the number of multichains of length k, that is chains with repeated entries, between x and y; in particular, ζ 2 (x, y) = |[x, y]| for all x ≤ y ∈ X.

Moreover, we have ∂ 1 n−1 n−1 X = 1+X ∈ Q[[X]], and going over to the associated n≥1 (−1) ∂X log = Taylor series we get exp(log) = 1 + X ∈ Q[[X]] and log(exp −1) = X ∈ Q[[X]]. If K is a field of characteristic 0, then for f ∈ XK[[X]] we have log(f ) := (−1)n−1 n f ∈ 1 + XK[[X]] ⊆ K[[X]], fulfilling the identity log((f + 1)(g + n≥0 n 1) − 1) = log(f ) + log(g) ∈ K[[X]], for all f, g ∈ XK[[X]]: We have exp(log((f + 1)(g+1)−1)) = (f +1)(g+1) = exp(log(f ))·exp(log(g)) = exp(log(f )+log(g)) ∈ K[[X]], thus log((f + 1)(g + 1) − 1) = log(exp(log((f + 1)(g + 1) − 1)) − 1) = log(exp(log(f ) + log(g)) − 1) = log(f ) + log(g) ∈ K[[X]].

Then for some k ∈ N we have i=1 λi ≤ j k k i=1 µi for all j ∈ {1, . . , k − 1}, and i=1 λi > i=1 µi . Hence we have n n n n λk > µk and i=k+1 λi < i=k+1 µi . Then we have i=k+1 λi = i=k+1 |{j ∈ N; i ≤ λj }| = λk j=1 (λj − k) and similarly λj ≥ k for j ∈ {1, . . , λk }, this implies µk j=1 (λj − k), thus µ λ, a contradiction. n i=k+1 µi = µk j=1 (µj − k) µk j=1 (µj − k). 8) Stratification of the nilpotent variety. a) Recall that C is equipped with the usual metric topology, that subsets of topological spaces are equipped with induced topologies, and that direct products of topological spaces are equipped with product topologies.

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Algebraic combinatorics by Jürgen Müller

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