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Full-Text Articles in Physical Sciences and Mathematics
A Combinatorial Miscellany: Antipodes, Parking Cars, And Descent Set Powers, Alexander Thomas Happ
A Combinatorial Miscellany: Antipodes, Parking Cars, And Descent Set Powers, Alexander Thomas Happ
Theses and Dissertations--Mathematics
In this dissertation we first introduce an extension of the notion of parking functions to cars of different sizes. We prove a product formula for the number of such sequences and provide a refinement using a multi-parameter extension of the Abel--Rothe polynomial. Next, we study the incidence Hopf algebra on the noncrossing partition lattice. We demonstrate a bijection between the terms in the canceled chain decomposition of its antipode and noncrossing hypertrees. Thirdly, we analyze the sum of the 𝑟th powers of the descent set statistic on permutations and how many small prime factors occur in these numbers. These results …
The Characteristic Polynomial Of The Adams Operators On Graded Connected Hopf Algebras, Marcelo Aguiar, Aaron Lauve
The Characteristic Polynomial Of The Adams Operators On Graded Connected Hopf Algebras, Marcelo Aguiar, Aaron Lauve
Mathematics and Statistics: Faculty Publications and Other Works
The Adams operators ‰n on a Hopf algebra H are the convolution powers of the identity of H. They are also called Hopf powers or Sweedler powers. We study the Adams operators when H is graded connected. The main result is a complete description of the characteristic polynomial — both eigenvalues and their multiplicities — for the action of the operator ‰n on each homogeneous component of H. The eigenvalues are powers of n. The multiplicities are independent of n, and in fact only depend on the dimension sequence of H. These results apply in particular to the antipode of …
An Alternative Approach To The Adem Relations In The Mod 2 Steenrod Algebra, Neşet Deni̇z Turgay
An Alternative Approach To The Adem Relations In The Mod 2 Steenrod Algebra, Neşet Deni̇z Turgay
Turkish Journal of Mathematics
The Leibniz--Hopf algebra F is the free associative algebra over Z on one generator S^n in each degree n>0, with coproduct given by \Delta(S^n) = \sum_{i+j=n} S^i \otimes S^j. We introduce a new perspective on the Adem relations in the mod 2 Steenrod algebra A_2 by studying the map \pi^\ast dual to the Hopf algebra epimorphism \pi: F \otimes Z/2 \to A_2. We also express Milnor's Hopf algebra conjugation formula in A_2^\ast in a different form and give a new approach for the conjugation invariant problem in A_2^\ast.