Lattice congruences, fans and Hopf algebras

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern-avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.Comment: 34 pages, 1 figur

Continuity argument revisited: geometry of root clustering via symmetric products

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as a symmetric product morphism. We describe the topology of strata up to homotopy equivalence and, for many important cases, up to homeomorphism. Adjacencies between strata are also described. Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.Comment: 45 pages, 4 figure

Which groups are amenable to proving exponent two for matrix multiplication?

The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $\omega$ in terms of the representation theory of the host group. This framework is general enough to capture the best known upper bounds on $\omega$ and is conjectured to be powerful enough to prove $\omega = 2$, although finding a suitable group and constructing such an embedding has remained elusive. Recently it was shown, by a generalization of the proof of the Cap Set Conjecture, that abelian groups of bounded exponent cannot prove $\omega = 2$ in this framework, which ruled out a family of potential constructions in the literature. In this paper we study nonabelian groups as potential hosts for an embedding. We prove two main results: (1) We show that a large class of nonabelian groups---nilpotent groups of bounded exponent satisfying a mild additional condition---cannot prove $\omega = 2$ in this framework. We do this by showing that the shrinkage rate of powers of the augmentation ideal is similar to the shrinkage rate of the number of functions over $(\mathbb{Z}/p\mathbb{Z})^n$ that are degree $d$ polynomials; our proof technique can be seen as a generalization of the polynomial method used to resolve the Cap Set Conjecture. (2) We show that symmetric groups $S_n$ cannot prove nontrivial bounds on $\omega$ when the embedding is via three Young subgroups---subgroups of the form $S_{k_1} \times S_{k_2} \times \dotsb \times S_{k_\ell}$---which is a natural strategy that includes all known constructions in $S_n$. By developing techniques for negative results in this paper, we hope to catalyze a fruitful interplay between the search for constructions proving bounds on $\omega$ and methods for ruling them out.Comment: 23 pages, 1 figur