The peak algebra and the descent algebras of types B and D
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Abstract
We show the existence of a unital subalgebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper P Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {P}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper P Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {P}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper P Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {P}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a two-sided ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove German upper P With ring Subscript n"> <mml:semantics> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mo> ∘ </mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\overset {\circ }{\mathfrak {P}}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is defined in terms of <italic>interior</italic> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 right-arrow ModifyingAbove German upper P With ring Subscript n Baseline right-arrow German upper P Subscript n Baseline right-arrow German upper P Subscript n minus 2 Baseline right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mo> ∘ </mml:mo> </mml:mover> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false"> → </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0\to \overset {\circ }{\mathfrak {P}}_n \to \mathfrak {P}_n\to \mathfrak {P}_{n-2}\to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left floor StartFraction n Over 2 EndFraction right floor plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⌊ </mml:mo> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo fence="false" stretchy="false"> ⌋ </mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\lfloor \frac {n}{2}\rfloor +1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper P Subscript n"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">P</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {P}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml">
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