By Y. He

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**Extra resources for Algebraic Singularities, Finite Graphs and D-Brane Theories**

**Example text**

Many interesting corollaries follow. Of the most useful are the following. Any representation R is irreducible iff (χR , χR ) = 1 and if not, then (χR , χRi ) gives the multiplicity of the decomposition of R into the i-th irrep. For the regular representation Rr , the character is simply χ(g) = 0 if g = II and it is |G| when g = II (this is simply because any group element h other than the identity will permute g ∈ G and in the vector basis eg correspond to a non-diagonal element and hence do not contribute to the trace).

LV g = 0) we can impose triholomorphicity: LV ωi = V ν (dωi )ν + d(V ν (ωi )ν ) = 0 which together with closedness dωi = 0 of the hyper-K¨ahler forms imply the existence of potentials µi , such that dµi = V ν (ωi )ν . , dim(G) These maps are the (hyper-K¨ahler) moment maps and usually grouped as µIR = µ3 and µC = µ1 + iµ2 Thus equipped, for any hyper-K¨ahler manifold Ξ of dimension 4n admitting k freely acting triholomorphic symmetries, we can construct another, Xζ , of dimension 4n − 4k by the following two steps: 1.

H)), we study the representation by associating vector spaces as follows: to each vertex q, we associate a pair of hermitian vector spaces Vq and Wq . We then define the complex vector space: r M(v, w) := h∈H := h,q Hom(Vα(h) , Vβ(h) ) ⊕ q=1 Hom(Wq , Vq ) ⊕ Hom(Vq , Wq ) {Bh , iq , jq } with v := (dimC V1 , ... , dimC Vn ) and w := (dimC W1 , ... , dimC Wn ) being vectors of dimensions of the spaces associated with the nodes. Upon M(v, w) we can introduce the action by a group G := q −1 U(Vq ) : {Bh , iq , jq } → gα(h) Bh gβ(h) , gq iq , jq gq−1 with each factor acting as the unitary group U(Vq ).

### Algebraic Singularities, Finite Graphs and D-Brane Theories by Y. He

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