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Register a new userRooted trees and symmetric functions: Zhaos homomorphism and the commutative hexagon
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Authors
Michael E. Hoffman
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Recent work on perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf algebras to each other and to the Hopf algebras of symmetric functions, noncommutative symmetric functions, and quasi-symmetric functions.
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The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations.
Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the moments F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a product of two matrices, ultimately yielding a polynomial in q=p^d. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric functions. This problem is closely to the study of maximal caps.
We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${mathbf K}^{mathfrak c}_n$. We show that ${mathbf K}^{mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${bf U}(hat{mathfrak{gl}}_n)$, and $big({mathbf U}(hat{ mathfrak{gl}}_n), {mathbf K}^{mathfrak c}_n)$ forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.
We continue study of some algebraic varieties (called resultantal varieties) started in a paper of A. Grishkov, D. Logachev Resultantal varieties related to zeroes of L-functions of Carlitz modules. These varieties are related with the Sylvester matrix for the resultant of two polynomials, from one side, and with the L-functions of twisted Carlitz modules, from another side. Surprisingly, these varieties are described in terms of finite weighted rooted binary trees. We give a (conjecturally) complete description of them, we find parametrizations of their irreducible components and their invariants: degrees, multiplicities, Jordan forms, Galois actions. Proof of the fact that this description is really complete is a subject of future research. Maybe a generalization of these results will give us a solution of the problem of boundedness of the analytic rank of twists of Carlitz modules.
We define and study a series indexed by rooted trees and with coefficients in Q(q). We show that it is related to a family of Lie idempotents. We prove that this series is a q-deformation of a more classical series and that some of its coefficients are Carlitz q-Bernoulli numbers.
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