Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications

Abstract

In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E 7 into a sum of irreducible submodules.Moreover, we obtain a combinatorial identity, saying that the dimensions of certain irreducible modules of E 7 are correlated by the binomial coefficients of fifty-five. Furthermore, we prove that two families of irreducible submodules with three integral parameters are solutions of the fundamental invariant differential operator corresponding to Cartan's unique quartic E 7 invariant.

Representations of Lie Algebras and Coding Theory

“…Furthermore, we prove that the codes C 3 (V ) of the exceptional simple Lie algebras F 4 , E 6 , E 7 and E 8 on their minimal irreducible modules and adjoint modules are all ternary orthogonal codes with large minimal distances. This coding theoretic phenomena was observed when we investigated the polynomial representations of these algebras in [26]- [28]. It is also well known that determining the minimal distance of a linear code is in general very difficult.…”

“…(5) The ternary weight code of E 7 on its minimal module is an orthogonal [28,7,12]code. (6) The ternary weight code of E 7 on its adjoint module is an orthogonal [63,7,27]-code.…”

“…) is a ternary orthogonal [10,4,6]-code when n = 5 (which is optimal (e.g., cf. [1])), [28,7,12] Finally, we consider the adjoint representation of sl(n, C). Note that {E i,j | 1 ≤ i < j ≤ n} are positive root vectors.…”

“…(6.18)DenoteA E 7 = (a r,i ) 7×28 . (6Theorem The ternary weight code C E 7 ,1 of E 7 on V E 7 is an orthogonal[28,7,12]code.…”

“…So (A E 7 , −A E 7 ) generates a ternary orthogonal [56, 7, 24]-code. Hence the ternary code C E 7 ,1 generated by A E 7 is an orthogonal[28,7,12]-code. ✷Next we consider the ternary weight code of E 7 on its adjoint module.…”