Let $g$ be an element in the permutation group (symmetric group) $S_N$. Define the Markov trace of $g$ (denoted $\text{tr}_k g$) as $$\text{tr}_kg = k^\text{number of cycles in $g$} ,$$ which depends on $k$. 1-cycles (fixed points) are also counted in the cycle number. For example, in $S_3$ group, $$\text{tr}_k(1)(2)(3)=k^3,\\\text{tr}_k(3)(12)=k^2,\\\text{tr}_k(132)=k.$$ This trace natually arise as the character of the permutation representation on the $N$-fold tensor power of a $k$-dimensional vector space if $k$ is a positive integer. But I would like to consider a more general case where $k$ is a positive real number.

The qustion is: whether there is a matrix representation $D_k(g)$ of $g$ such that the matrix trace is equal to the Markov trace? $$\text{Tr}D_k(g)=\text{tr}_k g.$$ If there exist such matrix representation, is there an explicit construction of the representation (an explicit method to calculate each matrix element $D_k(g)_{ij}$)?

realcharacter (defined using $k$, not $x$), meaning the thing is a real linear combination of actual characters. The question there was whether it was anonnegativereal linear combination of irreducible characters (this depends on the value of $k$). – David Handelman May 15 '16 at 13:27