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Register a new userInvariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables
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We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the place-action of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.
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Given a symmetric polynomial $P$ in $2n$ variables, there exists a unique symmetric polynomial $Q$ in $n$ variables such that [ P(x_1,ldots,x_n,x_1^{-1},ldots,x_n^{-1}) =Q(x_1+x_1^{-1},ldots,x_n+x_n^{-1}). ] We denote this polynomial $Q$ by $Phi_n(P)$ and show that $Phi_n$ is an epimorphism of algebras. We compute $Phi_n(P)$ for several families of symmetric polynomials $P$: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomials of the form $Phi_n(operatorname{s}_{lambda/mu}^{(2n)})$, where $operatorname{s}_{lambda/mu}^{(2n)}$ is a skew Schur polynomial in $2n$ variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as $x_1,ldots,x_n,x^{-1}_1,ldots,x^{-1}_n$. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.
In this note we give a version of Hao Huangs proof of the sensitivity conjecture, shedding some light on the origin of the magical matrix $A$ in that proof. For the history of the subject and the importance of this conjecture to the study of boolean functions, we refer to the original paper. Here we only state the main result: Consider the boolean cube $Q_n={0,1}^n$ as a graph, whose edges connect pairs of vertices differing in one coordinate. Then any its induced subgraph on greater than $2^{n-1}$ (the half) vertices has degree of some vertex at least $sqrt{n}$.
The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max, +), Izhakian-Rowens extended and supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. This allows us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.
We relate finite generation of cones, monoids, and ideals in increasing chains (the local situation) to equivariant finite generation of the corresponding limit objects (the global situation). For cones and monoids there is no analogue of Noetherianity as in the case of ideals and we demonstrate this in examples. As a remedy we find local-global correspondences for finite generation. These results are derived from a more general framework that relates finite generation under closure operations to equivariant finite generation under general families of maps. We also give a new proof that non-saturated Inc-invariant chains stabilize, closing a gap in the literature.
We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on $c$-clusters in the theory of cluster algebras.
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