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THE PEAK ALGEBRA THE DESCENT ALGEBRAS OF TYPES B AND D
 

Summary: THE PEAK ALGEBRA
AND
THE DESCENT ALGEBRAS OF TYPES B AND D
MARCELO AGUIAR, NANTEL BERGERON, AND KATHRYN NYMAN
Abstract. We show the existence of a unital subalgebra Pn 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 Pn 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 Pn contains a two-sided ideal

Pn
which is defined in terms of interior peaks. This object was introduced in previous work
by Nyman [28]; we find that it is the image of certain ideals of the descent algebras
of types B and D introduced in [4] and [9]. We derive an exact sequence of the form
0

Pn Pn Pn-2 0. We obtain this and many other properties of the peak
algebra and its peak ideal by first establishing analogous results for signed permuta-
tions and then forgetting the signs. In particular, we construct two new commutative
semisimple subalgebras of the descent algebra (of dimensions n and n

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics