Combinatorial Deformations of the Full Transformation Semigroup

We define two deformations of the Full Transformation Semigroup algebra. One makes the algebra “as semisimple as possible”, while another leads to an eigenvalue result involving Schur functions. Preliminaries The Full Transformation Semigroup on n letters, denoted Tn, is the semigroup of all set maps w : [n] → [n], where [n] = {1, 2, . . . , n} and the multiplication is the usual composition. Such maps can be depicted in several ways; we will most often use one-line notation, for example w = 214442 denotes the map sending 1 to 2, 2 to 1, 3 to 4, etc. Maps in Tn are indexed by triples (π, P, φ), where P is the image of the map, π is the set partition of [n] whose blocks are the inverse images of the elements of P , and φ is the permutation describing which block is mapped to which element of the image. In what follows, π = {π1, π2, . . .} will always denote a set partition of [n] with blocks ordered by increasing smallest element. Similarly in writing P = {p1, p2, . . .} a subset of [n] we shall always intend p1 < p2 < . . .. Permutations will be written in cycle notation. With these conventions, we shall let wπ,P,φ ∈ Tn denote the map taking x ∈ πi to pφ(i). Example 1. For w = 214442 ∈ T6 we have π(w) = 16|2|345, P (w) = {1, 2, 4}, and φ(w) = (12)(3), the transposition exchanging 1 and 2 and fixing 3. The permutation φ is most easily visualized in the following diagram of w. π: P : 1 2 4 16 2 345 @ @ The invertible elements of Tn, i.e., the bijective maps, form a subsemigroup isomorphic to the Symmetric Group Sn. Thus the elements of Tn can be thought of as generalized permutations, and we can ask which of the many combinatorial aspects of the Symmetric Group can be extended in a meaningful way to the Full Transformation Semigroup. Let CTn denote the Full Transformation Semigroup algebra, consisting of complex linear combinations of elements of Tn. CTn has a chain of two-sided ideals CTn = In ⊇ In−1 ⊇ . . . ⊇ I1 ⊇ I0 = 0, where for 1 ≤ k ≤ n, Ik as a vector space is the complex span of the maps of rank less than or equal to k (the rank of a map is the cardinality of its image). For