# Counting and Configurations by Jiri Herman, Radan Kucera, Jaromir Simsa By Jiri Herman, Radan Kucera, Jaromir Simsa

This booklet offers equipment of fixing difficulties in 3 components of ordinary combinatorial arithmetic: classical combinatorics, combinatorial mathematics, and combinatorial geometry. short theoretical discussions are instantly through rigorously worked-out examples of accelerating levels of hassle and via workouts that variety from regimen to relatively demanding. The ebook gains nearly 310 examples and 650 exercises.

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We can choose each f i in such a way that the isotropy subgroup W fi is a reﬂection subgroup of W of the form W Ii for some Ii ⊂ . For each 1 ≤ i ≤ m let W fi be a system of minimal length representatives of W/W fi as deﬁned above. Deﬁne m u −1 K ( ) × { f i } ⊂ T × Hom(A, T ) = X T . L( ) := i=1 u∈W fi Deﬁned in this way L( ) ⊂ X T is a sub CW-complex. We now show that L( ) is such that for every element x ∈ X T there is a unique w ∈ W such that wx ∈ L( ). To see this, since K ( ) ⊂ T satisﬁes this property, it sufﬁces to see that for any i and any v1 , v2 ∈ W there are unique v ∈ W and u ∈ W fi such that v1 K ( ) × v2 f i = v(u −1 K ( ) × f i ).

The same argument shows that the inclusion map i : G/T ×W F(π, f ) → G/T ×W Hom(π, G)1T1 f 19 On the structure of spaces of commuting elements in compact Lie groups induces a surjective homomorphism i ∗ : π1 (G/T ×W F(π, f )) → π1 (G/T ×W Hom(π, G)1T1 f ). 6]). 3. Suppose that G ∈ P and that π is a ﬁnitely generated abelian group. Then the map ϕ : G/T ×W Hom(π, G)1T1 f → Hom(π, G)11 f is π1 -surjective. Note that G/T ×W Hom(π, G)1T1 f ∼ = G/T ×W f T n . Since W f acts freely on G/T the projection map p induces a ﬁbration sequence p T n → G/T ×W f T n → (G/T )/W f ∼ = G/NG f (T ).

The tools applied in  can be 17 On the structure of spaces of commuting elements in compact Lie groups used to generalize this result to the class of spaces of homomorphisms Hom(π, G). Here we need to assume that π is a ﬁnitely generated abelian group and that G is a Lie group in the class P. 1] can be generalized for any choice of base point in Hom(π, G). Write π in the form π = Zn ⊕ A, where A is a ﬁnite abelian group. Suppose ﬁrst that n = 0 and thus π is a ﬁnite group. 5 there is a homeomorphism G/G f , : Hom(π, G) → [ f ]∈Hom(π,T )/W where each G f is a maximal rank isotropy subgroup with W (G f ) = W f .