The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V regarded as an FG-submodule of the free associative algebra (or tensor algebra) T(V). For each positive integer r, let L^r(V) and T^r(V) be the rth homogeneous components of L(V) and T(V), respectively. Here L^r(V) is called the rth Lie power of V. Our main result is that there are submodules B_1, B_2, ... of L(V) such that, for all r, B_r is a direct summand of T^r(V) and, whenever m \geq 0 and k is not divisible by p, $$L^{p^mk}(V) = L^{p^m}(B_k) \oplus L^{p^{m-1}}(B_{pk}) \oplus ... \oplus L^p(B_{p^{m-1}k}) \oplus L^1(B_{p^mk}).$$ Thus every Lie power is a direct sum of Lie powers of p-power degree. The approach builds on an analysis of T^r(V) as a bimodule for G and the Solomon descent algebra.Comment: 32 page

Teacher education and its association with decision-making: An investigation of the classroom management decisions of incoming education majors, graduating education majors, and expert teachers

This study investigates the value of a teacher education program by comparing freshman education majors and senior education majors in their ability to make decisions about classroom management issues. Participants (N = 137) responded to a vignette style interview schedule and responses were coded and analyzed. Senior education majors were found to make significantly better decisions than freshman education majors and two groups of non-education students. Implications for improving and evaluating teacher education are discussed.Educational Psycholog

Combinatorial Markov chains on linear extensions

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in terms of discrete time Markov chain

Supplementary material for the article: Nikolić, J.; Nešić, A.; Kull, S.; Schocker, F.; Jappe, U.; Gavrović-Jankulović, M. Employment of Proteomic and Immunological Based Methods for the Identification of Catalase as Novel Allergen from Banana. Journal of Proteomics 2018, 175, 87–94. https://doi.org/10.1016/j.jprot.2018.01.007

Supplementary material for: [https://doi.org/10.1016/j.jprot.2018.01.007]Related to published version: [http://cherry.chem.bg.ac.rs/handle/123456789/2117

Supplementary material for the article: Nikolić, J.; Nešić, A.; Kull, S.; Schocker, F.; Jappe, U.; Gavrović-Jankulović, M. Employment of Proteomic and Immunological Based Methods for the Identification of Catalase as Novel Allergen from Banana. Journal of Proteomics 2018, 175, 87–94. https://doi.org/10.1016/j.jprot.2018.01.007

Supplementary material for: [https://doi.org/10.1016/j.jprot.2018.01.007]Related to published version: [http://cherry.chem.bg.ac.rs/handle/123456789/2117