Row and column removal theorems for homomorphisms between Specht modules

Abstract

We prove analogues of (Donkin's generalisations of) James's row and column removal theorems, in the context of homomorphisms between Specht modules for symmetric groups

Homomorphisms between Specht modules

“…For d = 2, each of these coefficients is divisible by l r − s + 1 + γ r = 0, while for d = 1, each of the g(S) is divisible by l 0 + s − l 1 + m 1 − 1 − γ 1 = 6; so all the g(S) are divisible by 3. Hence Hom F 3 S 10 (S (7, 3) , S (4,3 2 ) ) = 0.…”

“…With ξ and ν as in Theorem 9, let Hom kS n (S ξ , S ν ) denote the space of homomorphisms from S ξ to S ν which are linear combinations of semi-standard homomorphisms. In the case where ξ and ν are obtained by removing the first column, Theorem 4.3 of [3] provides a linear injection from Hom(S ξ , S ν ) to Hom(S ξ , S ν ), which works even when the characteristic is two. Using [4,Theorem 8.15] we deduce a similar result for row removal.…”

Homomorphisms and Higher Extensions for Schur Algebras and Symmetric Groups

“…In the classical case a proof of this last result when m = 0 (given entirely in the context of the symmetric group) can be found in [FL03].…”

Homomorphisms between Weyl modules for $\operatorname {SL}_3(k)$

“…We get Θ (33,6) = {(33, 6), (34, 4), (33, 3), (31, 7), (30, 6), (31, 4), (28, 1), (19,13), (18,12), (19,10), (21, 9), (22,4), (24, 4)}.…”