Semisimple algebras related to immaculate tableaux

Abstract: Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity $n! = \sum_{\lambda \vdash n} ( f^{\lambda} )^{2}$ derived from the semisimple structure of the symmetric group algebra $\mathbb{C}S_{n}$, letting $f^{\lambda}$ denote the number of Young tableaux of partition shape $\lambda \vdash n$. By letting $g^{\alpha}$ denote the number of standard immaculate tableaux of composition shape $\alpha \vDash n$, we construct an algebra $\mathbb{C}\mathcal{I}{n}$ with a semisimple structure such that $\dim \mathbb{C}\mathcal{I}{n} = \sum_{\alpha \vDash n} (g^{\alpha})^{2}$ and such that $\mathbb{C}\mathcal{I}{n} $ contains an isomorphic copy of $\mathbb{C}S{n}$. We bijectively prove a recurrence for $\dim \mathbb{C}\mathcal{I}{n}$ so as to construct a basis of $\mathbb{C}\mathcal{I}{n}$ indexed by permutation-like objects that we refer to as immacutations. We form a basis $\mathcal{B}{n}$ of $\mathbb{C}\mathcal{I}{n}$ such that $\mathbb{C} \mathcal{B}_n$ has the structure of a monoid algebra in such a way so that $\mathcal{B}_n$ is closed under the multiplicative operation of $\mathbb{C} \mathcal{I}_n$, yielding a monoid structure on the set of order-$n$ immacutations.