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There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there are only two simple modules with restricted highest weights (abbreviated by non-negative integers), namely the trivial module $L(0)$ of dimension 1 and the natural module $L(1)$ of dimension 2. After this you need to rely on Steinberg's twisted tensor product theorem relative to a $p$-adic expansion of the highest weight. For instance, $L(2) \cong L(1)^{(1)}$, the first Frobenius twist of the natural module (still having dimension 2).

In your specific example, the recursion is easy to carry out: peel off the composition factor of highest weight and see what weights remain. Here yuu are looking at the tensor product of the natural module $L(1)^{(1)}$ with the "induced" module $H^0(3)$ of dimension 4 as in Jnntzen's book (polynomials in two variables of homogeneous degree 3) which is actually simple when $p=2$, isomorphic to $L(1) \otimes L(1)^{(1)}$. (These factors are the respective Steinberg modules for the first and second Frobenius kernels.)

From the recursion one arrives at a list of composition factors (having total dimension 8): $L(1)^{(2)}, \: L(1)^{(1)} \text{ twice }, L(0)\: \text{ twice}.$

Here at least there is a recursive method, but getting the precise module structure can be quite tricky. In this special case, you might take advantage of the fact that you are tensoring with a projective module for a certain Frobenius kernel. But in general it's complicated even in rank 1.

ADDED: For some recent work on decomposition of tensor products into indecomposables, see for example a paper by Doty and Henke, Decomposition of tensor products of modular irreducibles for SL$_2$. Q. J. Math. 56 (2005), no. 2, 189-207 (preprint here). This involves tilting modules and their Frobenius twists.

FURTHER COMMENTS: I intended to mention something about the extra question raised by Lloyd on invariant theory in prime characteristic. There is a classical result describing the ring of invariants for the general linear groups over a finite field, or for $\mathrm{SL}_n(\mathbb{F}_q)$, which goes back to L.E. Dickson in 1911. This has been reworked a number of times, for instance in an article by R. Steinberg, On Dickson’s theorem on invariants, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 699–707. (It's especially fitting to recall Steinberg's diverse contributions, since he died very recently on his 92nd birthday.)

In the special case dicussed here, Jantzen (unpublished) showed how to recover Dickson's theorem from the well-understood representation theory of the groups. This led me to investigate the general case in a similar spirit, though it remains to be seen whether we will know enough about representation theory to recover the full theorem this way. Anyway, my short paper contains an assortment of references to the literature (including work of Donkin on tilting modules and a paper by Wilkerson motivated by algebraic topology): Another look at Dickson's invariants for finite linear groups, Comm. Algebra 22 (1994), no. 12, 4773-4779.

There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there are only two simple modules with restricted highest weights (abbreviated by non-negative integers), namely the trivial module $L(0)$ of dimension 1 and the natural module $L(1)$ of dimension 2. After this you need to rely on Steinberg's twisted tensor product theorem relative to a $p$-adic expansion of the highest weight. For instance, $L(2) \cong L(1)^{(1)}$, the first Frobenius twist of the natural module (still having dimension 2).

In your specific example, the recursion is easy to carry out: peel off the composition factor of highest weight and see what weights remain. Here yuu are looking at the tensor product of the natural module $L(1)^{(1)}$ with the "induced" module $H^0(3)$ of dimension 4 as in Jnntzen's book (polynomials in two variables of homogeneous degree 3) which is actually simple when $p=2$, isomorphic to $L(1) \otimes L(1)^{(1)}$. (These factors are the respective Steinberg modules for the first and second Frobenius kernels.)

From the recursion one arrives at a list of composition factors (having total dimension 8): $L(1)^{(2)}, \: L(1)^{(1)} \text{ twice }, L(0)\: \text{ twice}.$

Here at least there is a recursive method, but getting the precise module structure can be quite tricky. In this special case, you might take advantage of the fact that you are tensoring with a projective module for a certain Frobenius kernel. But in general it's complicated even in rank 1.

ADDED: For some recent work on decomposition of tensor products into indecomposables, see for example a paper by Doty and Henke, Decomposition of tensor products of modular irreducibles for SL$_2$, Q. J. Math. 56 (2005), no. 2, 189-207 (arXiv:math/0205186). This involves tilting modules and their Frobenius twists.

FURTHER COMMENTS: I intended to mention something about the extra question raised by Lloyd on invariant theory in prime characteristic. There is a classical result describing the ring of invariants for the general linear groups over a finite field, or for $\mathrm{SL}_n(\mathbb{F}_q)$, which goes back to L.E. Dickson in 1911. This has been reworked a number of times, for instance in an article by R. Steinberg, On Dickson’s theorem on invariants, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 699–707. (It's especially fitting to recall Steinberg's diverse contributions, since he died very recently on his 92nd birthday.)

In the special case dicussed here, Jantzen (unpublished) showed how to recover Dickson's theorem from the well-understood representation theory of the groups. This led me to investigate the general case in a similar spirit, though it remains to be seen whether we will know enough about representation theory to recover the full theorem this way. Anyway, my short paper contains an assortment of references to the literature (including work of Donkin on tilting modules and a paper by Wilkerson motivated by algebraic topology): Another look at Dickson's invariants for finite linear groups, Comm. Algebra 22 (1994), no. 12, 4773-4779.

There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there are only two simple modules with restricted highest weights (abbreviated by non-negative integers), namely the trivial module $L(0)$ of dimension 1 and the natural module $L(1)$ of dimension 2. After this you need to rely on Steinberg's twisted tensor product theorem relative to a $p$-adic expansion of the highest weight. For instance, $L(2) \cong L(1)^{(1)}$, the first Frobenius twist of the natural module (still having dimension 2).

In your specific example, the recursion is easy to carry out: peel off the composition factor of highest weight and see what weights remain. Here yuu are looking at the tensor product of the natural module $L(1)^{(1)}$ with the "induced" module $H^0(3)$ of dimension 4 as in Jnntzen's book (polynomials in two variables of homogeneous degree 3) which is actually simple when $p=2$, isomorphic to $L(1) \otimes L(1)^{(1)}$. (These factors are the respective Steinberg modules for the first and second Frobenius kernels.)

From the recursion one arrives at a list of composition factors (having total dimension 8): $L(1)^{(2)}, \: L(1)^{(1)} \text{ twice }, L(0)\: \text{ twice}.$

Here at least there is a recursive method, but getting the precise module structure can be quite tricky. In this special case, you might take advantage of the fact that you are tensoring with a projective module for a certain Frobenius kernel. But in general it's complicated even in rank 1.

ADDED: For some recent work on decomposition of tensor products into indecomposables, see for example a paper by Doty and Henke, Decomposition of tensor products of modular irreducibles for SL$_2$. Q. J. Math. 56 (2005), no. 2, 189-207 (preprint here). This involves tilting modules and their Frobenius twists.

There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there are only two simple modules with restricted highest weights (abbreviated by non-negative integers), namely the trivial module $L(0)$ of dimension 1 and the natural module $L(1)$ of dimension 2. After this you need to rely on Steinberg's twisted tensor product theorem relative to a $p$-adic expansion of the highest weight. For instance, $L(2) \cong L(1)^{(1)}$, the first Frobenius twist of the natural module (still having dimension 2).

In your specific example, the recursion is easy to carry out: peel off the composition factor of highest weight and see what weights remain. Here yuu are looking at the tensor product of the natural module $L(1)^{(1)}$ with the "induced" module $H^0(3)$ of dimension 4 as in Jnntzen's book (polynomials in two variables of homogeneous degree 3) which is actually simple when $p=2$, isomorphic to $L(1) \otimes L(1)^{(1)}$. (These factors are the respective Steinberg modules for the first and second Frobenius kernels.)

From the recursion one arrives at a list of composition factors (having total dimension 8): $L(1)^{(2)}, \: L(1)^{(1)} \text{ twice }, L(0)\: \text{ twice}.$

Here at least there is a recursive method, but getting the precise module structure can be quite tricky. In this special case, you might take advantage of the fact that you are tensoring with a projective module for a certain Frobenius kernel. But in general it's complicated even in rank 1.

ADDED: For some recent work on decomposition of tensor products into indecomposables, see for example a paper by Doty and Henke, Decomposition of tensor products of modular irreducibles for SL$_2$. Q. J. Math. 56 (2005), no. 2, 189-207 (preprint here). This involves tilting modules and their Frobenius twists.

FURTHER COMMENTS: I intended to mention something about the extra question raised by Lloyd on invariant theory in prime characteristic. There is a classical result describing the ring of invariants for the general linear groups over a finite field, or for $\mathrm{SL}_n(\mathbb{F}_q)$, which goes back to L.E. Dickson in 1911. This has been reworked a number of times, for instance in an article by R. Steinberg, On Dickson’s theorem on invariants, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 699–707. (It's especially fitting to recall Steinberg's diverse contributions, since he died very recently on his 92nd birthday.)

In the special case dicussed here, Jantzen (unpublished) showed how to recover Dickson's theorem from the well-understood representation theory of the groups. This led me to investigate the general case in a similar spirit, though it remains to be seen whether we will know enough about representation theory to recover the full theorem this way. Anyway, my short paper contains an assortment of references to the literature (including work of Donkin on tilting modules and a paper by Wilkerson motivated by algebraic topology): Another look at Dickson's invariants for finite linear groups, Comm. Algebra 22 (1994), no. 12, 4773-4779.

There is unfortunately no "formiula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there are only t two simple modules with restricted highest weights (abbreivated by non-negative integers), namely the trivial module $L(0)$ of dimension 1 and the natural module $L(1)$ of dimension 2. After this you need to rely on Steinberg's ttwisted tensor product theorem relative to a $p$-adic expansion of the highest weight. For instance, $L(2) \cong L(1)^{(1)}$, the first Frobenius twist of the natural module (still having dimension 2).

In your specific example, the recursion is easy to carry out: peel off the composition factor of highest weight and see what weights remain. Here yuu are working with the tensor product of the natural module $L(1)^{(1)}$ with the "induced" module $H^0(3)$ of dimension 4 as in Jnntzen's book (polynomials in two variables of homogeneous degree 3) which is actually simple when $p=2$, isomorphic to $L(1) \otimes L(1)^{(1)}$. (These factors are the respective Steinberg modules for the first and second Frobenius kernels.)

From the recursion one arrives at a list of composition factors (having total dimension 8): $L(1)^{(2)}, \: L(1)^{(1)} \text{ twice }, L(0)\: \text{ twice}.$

Here at least there is a recrusive method, but getting the precise module structure can be quite tricky. In this special case, you might take advantage of the fact that you are tensoring with a projective module for a certain Frobenius kernel. But in general it's complicated even in rank 1.,

ADDED: For some recent work on decomposition of tensor products into indecomposables, see for example a paper by Doty and Henke, Decomposition of tensor products of modular irreducibles for SL$_2$. Q. J. Math. 56 (2005), no. 2, 189-207 (preprint here). This involves tilting modules and their Frobenius twists.

There is unfortunately no "formula" for tensor products in prime characteristic. Instead you can derive a list of composition factors $L(\lambda)$ (with multiplicity) by recursion. When $p=2$ there are only two simple modules with restricted highest weights (abbreviated by non-negative integers), namely the trivial module $L(0)$ of dimension 1 and the natural module $L(1)$ of dimension 2. After this you need to rely on Steinberg's twisted tensor product theorem relative to a $p$-adic expansion of the highest weight. For instance, $L(2) \cong L(1)^{(1)}$, the first Frobenius twist of the natural module (still having dimension 2).

In your specific example, the recursion is easy to carry out: peel off the composition factor of highest weight and see what weights remain. Here yuu are looking at the tensor product of the natural module $L(1)^{(1)}$ with the "induced" module $H^0(3)$ of dimension 4 as in Jnntzen's book (polynomials in two variables of homogeneous degree 3) which is actually simple when $p=2$, isomorphic to $L(1) \otimes L(1)^{(1)}$. (These factors are the respective Steinberg modules for the first and second Frobenius kernels.)

From the recursion one arrives at a list of composition factors (having total dimension 8): $L(1)^{(2)}, \: L(1)^{(1)} \text{ twice }, L(0)\: \text{ twice}.$

Here at least there is a recursive method, but getting the precise module structure can be quite tricky. In this special case, you might take advantage of the fact that you are tensoring with a projective module for a certain Frobenius kernel. But in general it's complicated even in rank 1.

ADDED: For some recent work on decomposition of tensor products into indecomposables, see for example a paper by Doty and Henke, Decomposition of tensor products of modular irreducibles for SL$_2$. Q. J. Math. 56 (2005), no. 2, 189-207 (preprint here). This involves tilting modules and their Frobenius twists.