(m)-Associahedra and (m)-Noncrossing Partition Polynomials and the Infinite Dihedral Group

Posted on July 26, 2023 by Tom Copeland

I was playing around once again yesterday with the basic algebraic relations among the sets of (m)-associahedra partition polynomials [A^{(m)}] and the sets of (m)-noncrossing partitions polynomials [N^{(m)}] = [N]^m, which I’ve presented in several posts over the last year or so, and decided to post a few defining relations for the dual sets of polynomials in a question on Math Stack Exchange in the hope that the group would be recognized by someone. The MSE user Karl pointed out that it sounds like I was describing the infinite dihedral group, linking to the associated Wikipedia article. This led me to the Wiki on the dihedral group and a set of group relations that are shared by my group of partition polynomials. Each set [N]^m plays the role of a rotation r_m in the dihedral group and each set [A^{(m)}], a reflection s_m. In the following I’ll show the applicability of these relations under the substitution operation I’ve illustrated in previous posts.

An infinite group \mathcal{N} is formed by iterating the substitution operation on [N]^1 = [N] and its inverse [N]^{-1}. The elements of this infinite group are [N]^{m} where m is any integer; [N]^0=[I], the identity; and, e.g., [N]^{3} = [N][N][N] and [N]^{-3} = [N]^{-1}[N]^{-1}[N]^{-1} under the repeated operation.

The infinite sets of (m)-associahedra partition polynomials satisfy for any integer m

(I)

[A^{(m)}]^2 = [I]

and

(II)

[N]^{\pm 1}[A^{(m)}] = [A^{(m\pm1)}].

That is, [A^{(m)}] is involutive and [N] and [N]^{-1} are the ladder ops–the raising and lowering ops–for the infinite set \mathcal{A} comprised of the infinite sets [A^{(m)}], where m runs over the infinite set of integers, as well as the ladder ops for the group \mathcal{N}.

For any integers i and j, clearly, by the definition of \mathcal{N} above,

[N]^i[N]^j = [N]^{i+j},

and the relation

[N]^{\pm 1}[A^{(m)}] = [A^{(m\pm1)}]

implies

[N]^{i}[A^{(j)}] = [A^{(j+i)}],

or equivalently

[N]^{-i}[A^{(j)}] = [A^{(j-i)}].

Recalling [A^{(m)}]^2 = [I], so [A^{(m)}]^{-1} =[A^{(m)}], and taking the inverse of this last equality gives

([N]^{-i}[A^{(j)}])^{-1} = ([A^{(j-i)}])^{-1},

implying

[A^{(j)}] [N]^{i} = [A^{(j-i)}].

Similarly, since

[A^{(j)}] = [N]^j[A^{(0)}] = [A^{(j)}]^{-1}= ([N]^j[A^{(0)}])^{-1} = [A^{(0)}] [N]^{-j},

finally

[A^{(i)}][A^{(j)}] = [N]^i[A^{(0)}][N]^j[A^{(0)}]

= [N]^i[A^{(0)}][A^{(0)}][N]^{-j}

= [N]^i[N]^{-j} = [N]^{i-j}.

Consequently, with [N]^i \to r_i and [A^{(i)}] \to s_i, my group satisfies the four relations

r_i r_j = r_{i+j}, \; r_i s_j = s_{i+j}, \; s_i r_j = s_{i-j}, \; s_i s_j = r_{i-j}

presented in the Wikipedia article on the dihedral group.

The group also satisfies the conjugation relations

1) [A^{(m)}][N]^n[A^{(m)}] = [N]^{-n}

and

2) [A^{(0)}][A^{(m)}][A^{(0)}] = [A^{(-m)}].

An analogous algebraic realization of the infinite dihedral group can be obtained by scaling the indeterminates by starting with multiplicative and compositional inversion of exponential generating functions (Taylor series) rather than ordinary generating function (power series). The extrapolation in both cases includes Laurent series. It would be interesting if there were other multivariate realizations under substitution.

Related stuff:

“Noncrossing partitions under rotation and reflection” by Callan and Smiley (pg.6)

“The power of group generators and relations: an examination of the concept and its applications” by Zhou

The Geometry and Topology of the Coxeter Groups by Davis

Group Theory by Milne

The CRM Winter School on Coxeter groups

Generalized Dihedral Group at GroupProps

“Dihedral group” by K. Conrad

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