Let $\lambda=(\lambda_1,\ldots,\lambda_k)$ and $\mu=(\mu_1,\ldots,\mu_r)$ be two strictly increasing sequences of non-negative integers ($r$ and $k$ may be $0$, in which case the sequence is empty, and the polynomial $f$ below is constant $1$). Define $$f(t)=\prod_{\substack{i=1,\ldots,k\\k=1,\ldots,r}}(t^{\lambda_i}+t^{\mu_j}).$$ Is there any canonical way to describe the degree of $f$ in terms of $\lambda$ and $\mu$?
Several cases are obvious, e.g. if the length of either $\lambda$ or $\mu$ is $\le 1$ then the degree is just the sum of the longer sequence. Just from this, it seems like it might be helpful to try and find a solution by considering $\lambda$ and $\mu$ as partitions, and trying to locate some formula using their duals.. but this is completely from intuition, I really have nothing to back this on..
Anyhow, I was wondering if perhaps someone here is familiar with this function and can help me with understanding it or with finding a relevant reference.
Thanks!
This polynomial $f$ appears in Lusztig's dimension formula for the degree of characters of finite classical groups of Lie-type (and that's also how I came about to consider it), so I'm adding the representation theory and finite group tags, just in case someone familiar might be acquainted with this.
Added: Perhaps it would be helpful to say something extra. My real interest is in the function $$\sum_{i=1}^k i\lambda_i+\sum_{j=1}^r j\mu_j+\deg(f).$$ Some theoretical ideas+checking cases with $r$ and $k$ small suggest that this last sum is dependent only in the values in the sequence $(\lambda_1,\ldots,\lambda_k,\mu_1,\ldots,\mu_r)$, and not on the way in which they are divided into $\lambda$ and $\mu$. I was wondering if this is known, or can be shown somehow that I can't quite see.