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We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step of understanding the notion of cluster superalgebra
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Richard Eager and Sebastian Franco introduced a change of basis transformation on the F-polynomials of Fomin and Zelevinsky, corresponding to rewriting them in the basis given by fractional brane charges rather than quiver gauge groups. This transformation seems to display a surprising stabilization property, apparently causing the first few terms of the polynomials at each step of the mutation sequence to coincide. Eager and Franco conjecture that this transformation will always cause the polynomials to converge to a formal power series as the number of mutations goes to infinity, at least for quivers possessing certain symmetries and along periodic mutation sequences respecting such symmetries. In this paper, we verify this convergence in the case of the Kronecker and Conifold quivers. We also investigate convergence in the $F_0$ quiver. We provide a combinatorial interpretation for the stable cluster variables in each appropriate case.
We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types $A_n$ or $widetilde{A}_n$ is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.
The goal of this paper is to define the Grassmann integral in terms of a limit of a sum around a well-defined contour so that Grassmann numbers gain geometric meaning rather than symbols. The unusual rescaling properties of the integration of an exponential is due to the fact that the integral attains the known values only over a specific set of contours and not over their rescale
The definitions of para-Grassmann variables and q-oscillator algebras are recalled. Some new properties are given. We then introduce appropriate coherent states as well as their dual states. This allows us to obtain a formula for the trace of a operator expressed as a function of the creation and annihilation operators.
We describe an infinite family of non-Plucker cluster variables inside the double Bruhat cell cluster algebra defined by Berenstein, Fomin, and Zelevinsky. These cluster variables occur in a family of subalgebras of the double Bruhat cell cluster algebra which we call Double Rim Hook (DRH) cluster algebras. We discover that all of the cluster variables are determinants of matrices of special form. We conjecture that all the cluster variables of the double Bruhat-cell cluster algebra have similar determinant form. We notice the resemblance between our staircase diagram and Auslander-Reiten quivers.
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