For a full exceptional sequence of vector bundles on the projective plane there is a remarkable equation, the so-called Markov equation, in terms of the ranks of the three vector bundles. This equation, slightly modified, has been used in a joint work with Beineke and Brustle for cluster mutations for quivers with three vertices. The aim of this talk is to define the natural generalization for full exceptional sequences with n members. This leads to the notion of a polynomial invariant, that is a polynomial in indeterminants x(i,j) for i < j between 1 and n. This allows to evaluate such a polynomial at any full exceptional sequence. We define a polynomial invariant to be a polynomial whose value does not depend on the full exceptional sequence, it only depends on the underlying category. In the talk we define polynomial invariants, present several examples and relate them to the natural braid group action. Eventually, we classify all polynomial invariants.

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Meeting ID: 9086116889
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Email: agnt@ipm.ir