On the commutative quotient of Fomin-Kirillov algebras

The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative algebra with a generator for each edge in the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, let $\mathcal E_G$ be the subalgebra of $\mathcal E_n$ generated by the edges in $G$. We show that the commutative quotient of $\mathcal E_G$ is isomorphic to the Orlik-Terao algebra of $G$. As a consequence, the Hilbert series of this quotient is given by $(-t)^n \chi_G(-t^{-1})$, where $\chi_G$ is the chromatic polynomial of $G$. We also give a reduction algorithm for the graded components of $\mathcal E_G$ that do not vanish in the commutative quotient and show that their structure is described by the combinatorics of noncrossing forests.Comment: 11 pages, 3 figure

Remarks on the non-commutative Khintchine inequalities for 0<p<20<p<2

We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to $R_k+C_k$ with $k\sim n^{\frac12}$. This naturally raises several interesting questions concerning the best possible such $k$. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for $0<p<1$ but we can prove several would be corollaries. For instance, given an infinite scalar matrix $[x_{ij}]$, we give a necessary and sufficient condition for $[\pm x_{ij}]$ to be in the Schatten class $S_p$ for almost all (independent) choices of signs $\pm 1$. We also characterize the bounded Schur multipliers from $S_2$ to $S_p$. The latter two characterizations extend to $0<p<1$ results already known for $1\le p\le2$. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting.Comment: Some more minor correction

Power series rings and projectivity

We show that a formal power series ring $A[[X]]$ over a noetherian ring $A$ is not a projective module unless $A$ is artinian. However, if $(A,{\mathfrak m})$ is local, then $A[[X]]$ behaves like a projective module in the sense that $Ext^p_A(A[[X]], M)=0$ for all ${\mathfrak m}$-adically complete $A$-modules. The latter result is shown more generally for any flat $A$-module $B$ instead of $A[[X]]$. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings.Comment: Mainly thanks to remarks and pointers by L.L.Avramov and S.Iyengar, we added further context and references. To appear in Manuscripta Mathematica. 7 page

Fractional Power Series and Pairings on Drinfeld Modules

Let $C$ be an algebraically closed field containing the finite field $F_q$ and complete with respect to an absolute value $|\;|$. We prove that under suitable constraints on the coefficients, the series $f(z) = \sum_{n \in \Z} a_n z^{q^n}$ converges to a surjective, open, continuous $F_q$-linear homomorphism $C \rightarrow C$ whose kernel is locally compact. We characterize the locally compact sub-$F_q$-vector spaces $G$ of $C$ which occur as kernels of such series, and describe the extent to which $G$ determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of $f$ of a given valuation, given the valuations of the coefficients. The ``adjoint'' series $f^\ast(z) = \sum_{n \in \Z} a_n^{1/q^n} z^{1/q^n}$ converges everywhere if and only if $f$ does, and in this case there is a natural bilinear pairing $$\ker f \times \ker f^\ast \rightarrow F_q$$ which exhibits $\ker f^\ast$ as the Pontryagin dual of $\ker f$. Many of these results extend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module