Abstract. To any Hamiltonian action of a reductive algebraic group G on a smooth irreducible symplectic variety X we associate certain combinatorial invariants: Cartan space, Weyl group, weight and root lattices. Actually, only the last definition is new. For cotangent bundles our invariants

Abstract. In this short note we show that Stanley–Reisner rings of simplicial complexes, which have had a “dramatic application ” in combinatorics [2, p. 41], possess a rigidity property in the sense that they determine their underlying simplicial complexes. For convenience we recall the notion of a Stanley–Reisner ring (for more information the reader is referred to [1, Ch. 5]). Let V be a finite set to be simplicial complex (on the vertex set V) if the following conditions hold: (a) {v} ∈ ∆ for any element v ∈ V, (b) σ ′ ∈ ∆ whenever σ ′ ⊂ σ for some σ ∈ ∆. Elements of ∆ will be called faces. Now assume we are given a field k and an abstract simplicial complex ∆ on a vertex set V. The Stanley–Reisner ring corresponding to these data is defined as the quotient ring of the polynomial ring k[v1,..., vn]/I, where n = #(V), the vi are the elements of V, and the ideal I is generated by the set of monomials {vi1 · · · vik |{vi1,..., vik} / ∈ ∆}. This k-algebra will be denoted by k[∆] and called the Stanley–Reisner ring of ∆. Further, the image of vi in it will again be denoted by vi (they are all different!) and hence will again be thought of as elements of V. Theorem. Let k be a field, and ∆ and ∆ ′ be two abstract simplicial complexes defined on the vertex sets V = {v1,..., vn} and U = {u1,..., um} respectively. Suppose k[∆] and k[ ∆ ′ ] are isomorphic as k-algebras. Then there exists a bijective mapping Ψ: V → U which induces an isomorphism between ∆ and ∆ ′. Proof. Let f: k[∆] → k[ ∆ ′ ] be a k-isomorphism. By scalar extension we may assume k is algebraically closed. Let us first show that without loss of

Presented approach in polynomial time calculates large number of invariants for each vertex, which won’t change with graph isomorphism and should fully determine the graph. For example numbers of closed paths of length k for given starting vertex, what can be though as the diagonal terms of k

a combinatorial invariant of the underlying simplicial complex, at least when there is no boundary. We also show how an extra phase factor arises when comparing Boltzmann weights under the Alexander moves, where the boundary undergoes subdivision. ITFA-93-06 / YCTP-P6-93

We prove that the rational homology Betti numbers of a toric variety with singularities are not necessarily determined by the combinatorial type of the fan which defines it; that is, the homology is not determined by the partially ordered set formed by the cones in the fan. We apply this result

Let K denote a closed odd-dimensional smooth manifold and let E be a flat vector bundle over K. In this situation the construction of Ray and Singer [RS] gives a metric on the determinant line of the cohomology det H ∗ (M; E) which is a smooth invariant of the manifold M and the flat bundle E

combinatorial objects related to Kazhdan-Lusztig and R-polynomials. The Bruhat graph of S n is the directed graph B(S n )

, by our homomorphism, we get a P(C)-valued invariant for M. We show that when applying to it the Bloch-Wigner function, it is zero. Under some conditions on M, we show the invariant lies in the Bloch group B(k), where k is the field generated by the cross-ratio. For a CR triangulation of Whitehead link

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