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The peak algebra and the descent algebras of types B and D


Authors: Marcelo Aguiar, Nantel Bergeron and Kathryn Nyman
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2781-2824
MSC (2000): Primary 05E99, 20F55; Secondary 05A99, 16W30
DOI: https://doi.org/10.1090/S0002-9947-04-03541-X
Published electronically: January 29, 2004
MathSciNet review: 2052597
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Abstract: We show the existence of a unital subalgebra $\mathfrak{P}_n$ of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that $\mathfrak{P}_n$ is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra $\mathfrak{P}_n$ contains a two-sided ideal $\overset{\circ}{\mathfrak{P}}_n$ which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form $0\to\overset{\circ}{\mathfrak{P}}_n \to\mathfrak{P}_n\to\mathfrak{P}_{n-2}\to 0$. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions $n$ and $\lfloor\frac{n}{2}\rfloor+1)$ by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces $\mathfrak{P}_n$ and $\overset{\circ}{\mathfrak{P}}_n$ over $n\geq 0$ and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.


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Additional Information

Marcelo Aguiar
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: maguiar@math.tamu.edu

Nantel Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
Email: bergeron@mathstat.yorku.ca

Kathryn Nyman
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: nyman@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03541-X
Keywords: Solomon's descent algebras, peak algebra, signed permutations, Coxeter groups, types B and D, Hopf algebras
Received by editor(s): March 3, 2003
Published electronically: January 29, 2004
Additional Notes: The first author thanks Swapneel Mahajan for sharing his insight on descent algebras and for interesting conversations.
The research of the second author was supported in part by CRC, NSERC and PREA
Article copyright: © Copyright 2004 American Mathematical Society

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