A LAnKe (also known as a Filippov algebra or a Lie algebra of the n-th kind) is a vector space equipped with a skew-symmetric n-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear part of the free LAnKe on 2n-1 generators as an irreducible representation. They announced that the multilinear component on 3n-2 generators decomposes as a direct sum of two irreducible symmetric group representations and a proof was given recently in a subsequent paper by Friedmann, Hanlon and Wachs. In the present paper we provide a proof of the later statement. The two proofs are substantially different.
A LAnKe (also known as a Lie algebra of the th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric -linear form that satisfies the generalized Jacobi identity. The symmetric group acts on the multilinear part of the free LAnKe on generators, where is the number of brackets, by permutation of the generators. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for , its irreducible decomposition contains no summand whose Young diagram has at most columns. The answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$
We analyze the impact of the institutional environment on the leverage of European listed SMEs for the period 2005-2018. We use a broad range of institutional quality, judicial efficiency and corruption measures, along with several firm-specific and macro control variables, to identify different transmission channels on leverage. By performing a panel data analysis into the fixed effects filter estimator framework, along with several model specifications and robustness tests, the results show that better institutions, stronger judicial effectiveness and higher corruption decrease leverage. In terms of active transmission channels, increased investment under regimes of better institutional quality tends to increase leverage. Higher judicial efficiency accompanied by increased profitability tends to decrease, while higher institutional quality accompanied by higher investments tends to increase leverage, bringing more bank credit. Increasing profitability under regimes of decreased corruption decreases leverage. This last finding is even more pronounced for medium enterprises, as opposed to micro enterprises. The most significant factors associated with leverage are profitability, asset structure, cost of borrowing, stock market development and size, while an age effect is rejected. Pecking order theory seems to better fit the European SMEs capital structure choices under several institutional states.
Given a filtration of the module of vector fields on a smooth manifold, we define a pseudodifferential calculus where the order of a vector field is given by the filtration. We show that pseudodifferential operators have a well-defined principal symbol for a subset of the unitary representations of the osculating groups. We prove a Rockland-type theorem, showing that the invertibility of the principal symbol is equivalent to maximal hypoellipticity. This answers affirmatively a conjecture due to Helffer and Nourrigat.