# what is an immanental polynomial

What is an immanental polynomial?

This comes up in a search, but I can't find a free version:
P. Botti and R. Merris, Almost all trees share a complete set of immanental polynomials, J. Graph Theory, 17 (1993) 467-476.

Is this a term they coined for the paper? I can't seem to find a definition anywhere.

If there is a good introduction to these polynomials, their properties, and applications, please do point it out.

• Not a commonly used term, and I don't have that book at hand to get context. But I have heard it used for certain generalizations of determinants (regarded as sums of matrix entries weighted by the character of representations of the symmetric group). here is a discussion. (to stress: I am not certain that this is the intended meaning, but it seems possible at least). – lulu Jul 30 '16 at 21:29

Just wrting down the definition from the paper you mentioned.

The determinant is only one of a class of matrix functions called immanants.

If $\chi$ is an irreducible character of the symmetric group $S_n$, and A = ($a_{ij}$) is an $n\times n$ matrix, then $$d_{\chi}(A) = \sum_{p \in S_n}\chi(p)\prod_{i=1}^{n}a_{ip(i)}$$

Note that the immanant corresponding to $\chi = \epsilon$, the alternating character, is the determinant; the immanant corresponding to , $\chi = 1$ is the permanent.

Now, because $\chi$ is a (conjugacy) class function, $d_{\chi}(P^{-1}AP) = d_{\chi}(A)$ for all $\chi$, for all A, and every n-by-n permutation matrix P.

Thus, $G_1$ and $G_2$ are isomorphic graphs only if they share a complete set of immanantal polynomials, i.e., only if $d_{\chi}(xI - A(G_1)) = d_{\chi}(xI - A(G_2))$ for all $\chi$.

• Are any of these other functions like the determinant in that P can be any unitary matrix instead of just a permutation matrix? It's weird to think of a generalization of a "characteristic polynomial", that isn't preserved when changing basis for a matrix. – JustThinking Jul 31 '16 at 0:38