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Weyl's dimension formula (for semisimple groups) shows immediately that the irreducible representation with highest weight $\rho$ has the dimension you state: $2^k$ with $k$ the number of positive roo …

The foundations seem to come from the book Representation Theory and Complex Geometry (Birkhauser, 1997) by Victor Ginzburg (with his former student Chriss), though in the question here you rely mainl …

Unlike the well-known case of these groups over a prime field, it's probably asking too much to exhibit full character tables over all such finite rings. (Note too that Kutzko limits his discussion …

To expand my short comment, I think it's possible to say something at least qualitative about the existence question you've raised. This goes back to Brauer's early work, some done with his student …

It's not difficult to answer this question, but for this it's useful to sketch briefly the origins of the term Weyl module. As usual in mathematics, the history and attributions are somewhat convolu …

This is meant as an extended comment (sometimes correction) to things said in various answers and comments. As a public service I'll try to fill in some of the history and references. I recall some …

This example is elementary (close to an exercise), which does make it tempting just to write a couple of lines of explanation in a paper. But including such a "proof" without further comment tends t …

For a more elementary approach, avoiding induction and Mackey theory, you might try a concrete construction. Realize Aakumadula's group $G$ as a $2 \times 2$ matrix group over $\mathbb{F}_p$ (say fo …

As Evan points out, "modern" technology (including Littelmann paths and canonical bases) provides an improved way to think about tensor product decompositions for simple Lie algebras. But your Theor …

The essential work in this direction was published from 1994 on by J.-P. Serre and J.C. Jantzen, concerning both algebraic groups and related finite groups of Lie type. Related papers by R. Guralnick …

This is an edited version of an earlier hasty try, which went off the tracks and was removed. As both Xandi Tuni and Someone have made clear in their comments, the answer to the stated question is " …

Like others I am skeptical about the possibility of finding an elementary proof of the dimension formula which is detached completely from the character formula. But of course I can't prove an imposs …

Here is at least a partial answer to the question, to supplement some comments I already made. The essential case is that of a simple Lie algebra over $\mathbb{C}$. For each simple type there is a …

The answers given by Bruce and Anton are sensible, but I'd add that it's important in Lie theory to distinguish the behavior of reductive groups from that of arbitrary Lie groups: finite dimensional r …

The question is not well-formulated, as Bugs Bunny observes. In the case of the special linear group or Lie algebra (say over the field $\mathbb{C}$), it happens to be true that the fundamental domi …