# The Representation of $\mathrm{Sp}_{2n}$ of Dimension $2^n$ in characteristic 2

Let $G:=\mathrm{Sp}_{2n}$ be the simple algebraic group of simply connected type with root-system $C_n$. Is there a way, to explicitly construct the highest weight representation $\mathrm{L}(\lambda)$, where $\lambda \in X(T)$ is the fundamental dominant weight corresponding to the shortest root in the Basis?

This should be the highest weight module of dimension $\dim{\mathrm{L}(\lambda)}=2^n$, where $n$ is the rank of the group.

So I'm looking for a construction of the corresponding matrix-representation of the finte group $G(\mathbb{F}_2)$, which could be implemented in an computer-algebra system (e.g. GAP) for arbitrary $n \geq 2$.

I need these matrices in particular for $n\geq 7$ for some computations. For dimension up to $12$ I have got the other representations as composition factors of tensor products of the natural representation with the Meat-Axe.

• In positive characteristic it is generally difficult to determine the dimension of higher-weight representations (could be smaller than in characteristic 0); how do you know for that particular $\lambda$ that the dimension is $2^n$ for characteristic 2? – Marguax Sep 28 '13 at 13:44
• There is this paper from Frank Lübeck link and I have computed the composition factors of the tensorproducts of the natural module. If $n \leq 6$, I got the desired representation this way. however, if $n \geq 7$, the dimensions get to big, to compute the factors. I think, the representations I'm looking for correspond somehow, becaus of the duality in characteristic $2$ of $B_n$ and $C_n$, to Spin-representations of the orthogonal groups $\mathrm{SO}_{2n+1}$ wich are isomorphic to $\mathrm{Sp}_{2n}$ in even char.. – KoopaTroopa Sep 28 '13 at 14:16
• SO$_{2n+1}$ and Sp$_{2n}$ are not isomorphic as algebraic groups in characteristic 2 (the natural isogeny between them in characteristic 2 has nontrivial unipotent infinitesimal kernel). Their groups of rational points (over a perfect field) are identified via that isogeny, but that is weaker than algebraic isomorphism. – Marguax Sep 28 '13 at 15:01
• I'm sorry, I should have written isomorphic as groups or the corresponding finite groups of Lie-Type over $\mathbb{F}_2$ are isomorphic. – KoopaTroopa Oct 4 '13 at 15:43

## 1 Answer

The basic theory here (for $p=2$) was worked out by R. Steinberg in his 1963 Nagoya Math. J. paper on representations of algebraic groups. On the other hand, computing explicit matrices for rank at least 7 promises to be cumbersome, unless you limit attention to key operators in the group corresponding to roots, etc. (Frank Lubeck and his collaborators have probably done the most sophisticated computing, but usually with a focus on basic root and weight data.)

Your formulation needs a little more care. It's true here that the simply connected algebraic groups of types $B_n, C_n$ are related in characteristic 2 by one of Chevalley's special isogenies; this in turn makes the underlying abstract groups over an algebraically closed field, or a finite field, isomorphic. Steinberg's general study of irreducible representations shows that these are realized as "twisted" tensor products of a basic collection belonging to the finite group over the prime field (or a corresponding restricted Lie algebra). So it's possible to study the representations of interest to you by working only with the finite groups over $\mathbb{F}_2$.

Steinberg's ideas are explained (carefully, I hope) in Sections 5.3-5.4 of my LMS Lecture Note volume Modular Representations of Finite Groups of Lie Type (2006). In particular, the theory leads to collections of modules supported on only long or only short roots in a precise way. This is illustrated at the end of my 5.4 for the groups $B_3, C_3$ and their three fundamental weights. The dimensions of irreducibles coincide and are respectively $6, 14, 8$. In general you do get $2^n$ for the last one, coming essentially from type $B_n$ (where there is a unique short simple root numbered $\alpha_n$). .

Short of doing computer calculations for such groups, there is some useful literature summarized in Section 4.5 of my book. This is due to Premet-Suprunenko, McNinch, and Foulle in particular.

P.S. I should add that the fundamental weight for $B_n$ which gives rise to the dimension $2^n$ is minuscule, so there is a single orbit of weights under the Weyl group. In particular, a basis for the representation space is uniquely determined up to scalars, though it's still so big that explicit matrices become almost impossible to write down as $n$ increases.