Relations in the Grothendieck group Sample Clauses

Relations in the Grothendieck group. In order to prove some results towards Conjecture 2.2, we study relations in the Grothendieck group of finitely generated Fp[GL2(Fv)] modules, just like in Chapter 2. Recall that as in [DR15] we write R ≤ R′ whenever R′ — R is in the submonoid of G0(Fp[GL2(Fv)]) consisting of classes of Fp-representations of GL2(Fv). If σ, σ′ are Fp-representations of GL2(Fv) and if σ is irreducible, then [σ] ≤ [σ′] if and only if σ is a Jordan–H¨older factor of σ′. So the Grothendieck group enables us to determine the Jordan–H¨older factors of the representations we are interested in. For k < —1, we define [Symk] := —[detk+1 ⊗ Sym−k−2], and we set [Sym−1] := 0. [f−i] For any k ∈ Z and Symk, we write Symk for its ith Frobenius twist. We i+1 choose this convention so that it aligns with our convention that τp = τi for embeddings in Σv. In the following we will often omit the [·] when it is clear we are working in the Grothendieck group. 1. Suppose fv > 1. We have the following relations in G0(Fp[GL2(Fv)]): (1) Let m, n ≥ 0. Then n+1 m+p [i] (3.1) detpi+1 ⊗ Symn ⊗ Symm ≤ Sym ⊗ Sym . (2) Suppose np > m ≥ 0, m = m0 + rp with 0 ≤ m0 ≤ p — 1. Then Symn−1 ⊗ Symm+p = Symn ⊗ Symm + detp (r+1) ⊗ Symn−r−2 ⊗ Symm0 + detpi(m+1) ⊗ Symn−r−1 ⊗ Symp−2−m0 .