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I know that in an old paper, R. Steinberg computed the irreducible (complex) characters of groups $GL_n(q)$ and $PGL_n(q)$ where $n \in$ {3, 4}. I want to know is there any known method for computing irreducible "character degrees", not the whole character table, of groups $GL_n(q)$ for arbitrary $n$ and $q$? (Here by character I mean the ordinary character).

If there isn't a general method, for which other pairs of $(n, q)$ the degree of irreducible characters of above groups are known?

Update Dec 23th: In his celebrated paper The Characters of Finite General linear Groups, J.A. Green has shown how to calculate character tables of $GL(n, q)$. It is shown that the complete character table could be calculated as soon as the values of certain polynomials $Q_{\mathcal{P}}^{\lambda}(q)$, which have been defined for each pair $(\mathcal{P})=(1^{z_1}, 2^{z_2},..., n^{z_n})$ and $(\lambda)=(\lambda_1,..., \lambda_k)$ of partitions of the number $n$, are known.

Green computed $Q_{\mathcal{P}}^{\lambda}(q)$ for $n=1,2,3,4,5$ by a recursive method. Also the values of $Q_{\mathcal{P}}^{\lambda}(q)$ for $n=6, 7$ are calculated in A. O. Morris paper The Characters of Group $GL(n, q)$. Here I am interested in computational aspects of the problem rather than just theoretical description of characters:

My Main Questions

1- What are further developements in computing the values of $Q_{\mathcal{P}}^{\lambda}(q)$?

2- If we just consider the degree of irreducible characters, not constructing the entire table, is there any method (that preferably doesn't need complex calculations of $Q_{\mathcal{P}}^{\lambda}(q)$) for calculating the "character degrees" of finite general linear groups?

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  • $\begingroup$ As far as I know, the entire character table is known; see en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory. $\endgroup$ – Qiaochu Yuan Dec 22 '15 at 21:21
  • $\begingroup$ @QiaochuYuan As far as I know, on finite general linear groups, Deligne-Lusztig theory is actually the same as what J. A. Green has done in his 1955 paper "characters of finite general linear groups". In that paper, Green use concepts of Hall polynomials and Symmetric functions to "describle" characters of $GL(n, q)$. Since I'm not much familiar with those concepts and I just need the "character degrees", I wonder whether or not there is a simpler approach to compute just the degrees? $\endgroup$ – user97635 Dec 22 '15 at 22:30
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I think reading the original paper of Green, which you have mentioned, is a good way of learning about characters of general linear groups. It's quite old and It doesn't need so much background. But If you are not comfortable with its old style or having difficulties with concepts of symmetric functions and Hall polynomials, then I. G. Macdonald's book Symmetric Functions and Hall Polynomials would be a good source to refer to [see Chapter IV: The Characters of $GL_n(q)$ Over A Finite Field, which discuss the Green's paper].

However If you just care about the $\textsf{degrees}$ of irreducible characters, there exists a database for irreducible character degrees (and their multiplicities) of many finite groups of lie type of rank at most $9$, include $PGL_{l+1}(q)$, $l \leq 8$.
These data are available in Frank Lübeck's homepage.

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