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hello,

Let $\lambda=(\lambda_1,\ldots ,\lambda_k)$ be a partition of $n$ and let $\chi_{\lambda}$ be the corresponding irreducible character of the symmetric group $S_n$.

The immanant of a matrix $A\in M_n(\mathbb{C})$ is defined to be

$$Imm_{\lambda}(A)=\sum_{\sigma\in S_n}\chi_{\lambda}(\sigma)a_{1\sigma(1)}\ldots a_{n\sigma(n)}.$$

The determinant is the special case in which the character is the alternating character and the permanent the case where the character is the principal character. both, the determinant and the permanent, can be defined naturally in a field of positive characteristic $p$. My question is: how can we generalize the remaining immanants to a field of positive characteristic $p$?

thank you

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    $\begingroup$ Characters of the symmetric group are always integer-valued. What's the difficulty? $\endgroup$
    – Qiaochu Yuan
    Apr 2, 2012 at 0:02
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    $\begingroup$ Even if the characters weren't ${\bf Z}$-valued, they'd have to be integers in some number field (a cyclotomic one, even), so one could reduce modulo a prime of that field. $\endgroup$
    – Noam D. Elkies
    Apr 2, 2012 at 1:13

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