NDLI: New results on the peak algebra

Content Provider	Springer Nature Link
Author	Aguiar, Marcelo Nyman, Kathryn Orellana, Rosa
Copyright Year	2006
Abstract	The peak algebra $$\mathfrak{P}_{n}$$ is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of $$\mathfrak{P}_{n}$$ . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of $$\mathfrak{P}_{n}$$ and to characterize the elements of $$\mathfrak{P}_{n}$$ in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals $$\mathfrak{P}_{n}^{j}$$ of $$\mathfrak{P}_{n}$$ , j = 0,..., $${\lfloor \frac{n}{2}\rfloor}$$ , such that $$\mathfrak{P}_{n}^{0}$$ is the linear span of sums of permutations with a common set of interior peaks and $$\smash{\mathfrak{P}_{n}{\lfloor \frac{n}{2}\rfloor}}$$ is the peak algebra. We extend the above results to $$\mathfrak{P}_{n}^{j}$$ , generalizing results of Schocker (the case j = 0).
Starting Page	149
Ending Page	188
Page Count	40
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	23
Issue Number	2
e-ISSN	15729192
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2006-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Solomon's descent algebra Peak algebra Signed permutation Type B Eulerian idempotent Free Lie algebra Jacobson radical Computer Science Group Theory and Generalizations Order, Lattices, Ordered Algebraic Structures Combinatorics Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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