Edge partitions of the complete graph and a determinant-like function

Existence of the detS2$det^{S^2}$ map

“…□ Remark 4.3. The result in this section is consistent with conjecture from [2] where it was proposed that the 𝑑𝑒𝑡 𝑆 2 map can be written as a sum over the set of homogeneous cycle free 𝑑-partition of the graph…”

“…Take 𝐷 ∶ 𝑉 ⊗𝑑(2𝑑−1) 𝑑 → 𝑘 to be a linear map with the property that 𝐷(⊗ 1⩽𝑖<𝑗⩽2𝑑 (𝑣 𝑖,𝑗 )) = 0 if there exist 1 ⩽ 𝑥 < 𝑦 < 𝑧 ⩽ 2𝑑 such 𝑣 𝑥,𝑦 = 𝑣 𝑥,𝑧 = 𝑣 𝑦,𝑧 . It follows from results proved in [2] (see Lemma 2.4), that if the 𝑑-partition (Γ 1 , … , Γ 𝑑 ) is not cycle-free, then 𝐷(𝑓 (Γ 1 ,…,Γ 𝑑 ) ) = 0. However, this exact statement was not made explicitly in [2], so for completeness sake, we recall the main steps of that proof.…”

“…Next we recall from [2,3] and [4] a few results about Λ 𝑆 2 𝑉 𝑑 and the 𝑑𝑒𝑡 𝑆 2 map. For every 𝑛 ⩾ 0 we define…”

“…Let  𝑑 = {𝑒 1 , 𝑒 2 , … , 𝑒 𝑑 } be a basis for the vector space 𝑉 𝑑 . It was noticed in [2] that the set…”

“…If the length of the cycle in Γ 𝑘 is 𝑙 ⩾ 4, then it was shown in [2] (see Lemma 3.5), that there exist two 𝑑-partition (Γ (1) 1 , … , Γ (1) 𝑑 ) and (Γ (2) 1 , … , Γ (2) 𝑑 ) of 𝐾 2𝑑 such that Γ (1) 𝑘 and Γ (2) 𝑘 have cycles of length 𝑙 − 1, and…”