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Register a new userIntroducing supersymmetric frieze patterns and linear difference operators
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We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hills operators. The space of these superfriezes is an algebraic supervariety, which is isomorphic to the space of supersymmetric second order difference equations, called Hills equations.
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We introduce a new class of friezes which is related to symplectic geometry. On the algebraic and combinatrics sides, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of type ${rm C}_{2}$ and ${rm A}_{m}$. On the geometric side, they are related to the moduli space of Lagrangian configurations of points in the 4-dimensional symplectic space introduced in [Conley C.H., Ovsienko V., Math. Ann. 375 (2019), 1105-1145]. Symplectic friezes share similar combinatorial properties to those of Coxeter friezes and SL-friezes.
Divided difference operators are degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the equivariant cohomology of G/P for arbitrary partial flag varieties of arbitrary Lie type, and show how to use them in the ordinary cohomology of G/P. We provide three applications. The first shows that all Schubert classes of partial flag varieties can be generated from a sequence of divided difference operators on the highest-degree Schubert class. The second is a generalization of Billeys formula for the localizations of equivariant Schubert classes of flag varieties to arbitrary partial flag varieties. The third gives a choice of Schubert polynomials for partial flag varieties as well as an explicit formula for each. We focus on the example of maximal Grassmannians, including Grassmannians of k-planes in a complex n-dimensional vector space.
An important instance of Rota-Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota-Baxter operator and how this is related to the ordered monoid.
Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix $A$ is an $ntimes n$ pseudo-involution then the singular values of $A$ must come in reciprocal pairs in $Sigma$ of a singular value decomposition $A=USigma V^T$. Moreover, we give a complete analysis of the existence and nonexistence of eigenvectors of Riordan matrices. As a result, we obtain a surprising partition of the group of Riordan matrices into three different types of eigenvectors. Finally, given a nonzero vector $v$, we investigate the algebraic structure of Riordan matrices $A$ that stabilize the vector $v$, i.e. $Av=v$.
Frieze patterns of numbers, introduced in the early 70s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present paper aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.
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