[Paper Review] The Flag Descent Algebra and the Colored Eulerian Descent Algebra

Matthew Moynihan|arXiv (Cornell University)|Oct 15, 2012

Advanced Combinatorial Mathematics21 references3 citations

TL;DR

This paper introduces and characterizes the flag descent algebra and the colored Eulerian descent algebra within the group algebra of the hyperoctahedral and colored permutation groups, respectively. It proves the existence of these subalgebras using colored $P$-partitions and constructs orthogonal idempotents that generalize the classical Eulerian idempotents, showing that the type $A$ Eulerian descent algebra is a two-sided ideal in the flag descent algebra.

ABSTRACT

We prove that the group algebra of the hyperoctahedral group contains a subalgebra corresponding to the flag descent number of Adin, Brenti, and Roichman. This algebra is in fact the span of the basis elements of the type A and type B Eulerian descent algebras. We describe a set of orthogonal idempotents which spans the flag descent algebra and prove that it contains the type A Eulerian descent algebra as a two-sided ideal. Using a new colored analogue of Stanley's $P$-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algebra. We also describe a set of orthogonal idempotents that spans the colored Eulerian descent algebra and includes, as a special case, the familiar Eulerian idempotents in the group algebra of the symmetric group.

Motivation & Objective

• To establish the existence of a flag descent algebra in the group algebra of the hyperoctahedral group using the flag descent statistic.
• To define and prove the existence of a colored Eulerian descent algebra in the Mantaci-Reutenauer algebra via a new colored analogue of Stanley’s $P$-partitions.
• To construct a set of orthogonal idempotents that span the colored Eulerian descent algebra, reducing to classical Eulerian idempotents when $r=1$.
• To show that the type $A$ Eulerian descent algebra is a two-sided ideal in the flag descent algebra, extending known results.

Proposed method

• Uses a new colored analogue of Stanley’s $P$-partitions to define descent statistics for colored permutations.
• Applies multiset enumeration techniques and generating functions to analyze descent and major index statistics.
• Employs signed and augmented $P$-partitions, including Chow’s and Petersen’s variants, to extend results from symmetric to hyperoctahedral groups.
• Derives a functional equation $\phi(x)\phi(y) = \phi(rxy + x + y)$ in the group algebra to construct orthogonal idempotents.
• Uses coefficient comparison in generating functions to prove that the map $\phi$ satisfies a multiplicative identity, enabling idempotent construction.
• Analyzes set partitions induced by descent statistics and proves that only the descent number with fixed boundary values induces a subalgebra.

Experimental results

Research questions

• RQ1 Does the flag descent number of Adin, Brenti, and Roichman define a subalgebra in the group algebra of the hyperoctahedral group?
• RQ2 Can a colored Eulerian descent algebra be defined in the Mantaci-Reutenauer algebra using Steingrímsson’s descent definition?
• RQ3 Are there orthogonal idempotents in the colored Eulerian descent algebra that reduce to the classical Eulerian idempotents when $r=1$?
• RQ4 Is the type $A$ Eulerian descent algebra a two-sided ideal in the flag descent algebra?
• RQ5 Which descent statistics induce a subalgebra structure in the colored permutation group algebra?

Key findings

• The flag descent algebra exists and is spanned by basis elements from both type $A$ and type $B$ Eulerian descent algebras.
• The type $A$ Eulerian descent algebra is a two-sided ideal in the flag descent algebra, extending a result of Petersen.
• A functional equation $\phi(x)\phi(y) = \phi(rxy + x + y)$ holds in the group algebra, enabling the construction of orthogonal idempotents.
• The orthogonal idempotents $c_i$ in the colored Eulerian descent algebra are explicitly computed for $r=5, n=3$, with rational coefficients in terms of descent class sums.
• Only the descent number with fixed boundary values $\pi(0) = 0_a, \pi(n+1) = 0_b$ induces a subalgebra; other boundary choices fail to close under multiplication.
• The descent set statistic does not induce a subalgebra in $G_{2,2}$, as products of basis elements cannot be expressed as linear combinations of descent set basis elements.