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Congruence Subgroups and Generalized Frobenius-Schur Indicators 37 c = dim A(U , U ⊗ U ) of A and s are related by The fusion coefficients Nab c a b Verlinde’s formula (cf. [BK01]): c = Nab d∈ sad sbd scd . 4) c , the assignment K (A) → Defining the matrix Na ∈ M (k) by (Na )bc = Nab 0 M (Z); [Ua ] → Na is the regular representation of K0 (A) in matrix form. 5) s where Da is the diagonal matrix [δi j s0a jj ] . 1. 3). For J, K ∈ GL( , k), we define Ya (J, K ) ∈ M (k) with the (b, c)-entry given by c Yab (J, K ) := d∈ sad Q bd Q cd , s0d where Q = J s K s J = [Q i j ] and Q −1 = [Q i j ] .

Proof. The involution σ1 simply exchanges the colors x and y of all graphs which induces the involution τ1 on tder2 . The involution σ2 flips the sign of the one form dφe for each edge (since the reflection changes sign of the Euclidean angle), and changes the orientation of each integration over a complex variable. Hence, for a graph with n internal vertices we collect −1 to the power (2n +1)+n ≡ n +1 (mod 2). Corresponding rooted trees have exactly n +1 leaves. Hence, one should change a sign of each leaf which results in applying the involution τ2 .

X j , 0, . . , xi , . . , 0) with x j placed at the position i and xi at the position j. Next, for the first q points collapsing inside the upper half plane, we have a stratum of the form Cq × Cn−q+1,0 . ,n . Other choices of points to collapse can be described by using the action of the symmetric group Sn . ,n . ,n |Cn−q,1 . Note that the restriction of the connection form ωn to the boundary stratum Cn−1,1 corresponds to configurations with the point z 1 on the real axis, and it has the following property: its first component (as an element of tdern ) vanishes since the 1-form dφe vanishes when the source of the edge e is bound to the real axis.