A graph-theoretic approach to a conjecture of Dixon and Pressman Matthew Brassil and Zinovy Reichstein Abstract. Given n × n matrices, A1,...,Ak, consider the linear operator L(A1,...,Ak): Matn → Matn given by L(A1,...,Ak)(Ak )= Pσ S sgn(σ)Aσ Aσ ··· Aσ k . The Amitsur- +1 ∈ k+1 (1) (2) ( +1) Levitzki theorem asserts that L(A1,...,Ak) is identically 0 for every k > 2n − 1. Dixon and Pressman conjectured that if k is an even number between 2 and 2n − 2, then the kernel of L(A1,...,Ak) is of dimension k for A1,...,Ak ∈ Matn(R) in general position. We prove this conjecture using graph-theoretic techniques. 1. Introduction Recall that the standard polynomial [A1,...,Am] in m variables is defined as [A1,...,Am]= sgn(σ)Aσ(1) ··· Aσ(m) . σX∈Sm The celebrated theorem of Amitsur and Levitzki [1] asserts that [A1,...,Am] = 0 for any m > 2n and any n × n-matrices A1,...,Am ∈ Matn(Λ) over a commutative ring Λ. The original proof in [1] is quite involved. Simpler proofs have since been given by Swan [7, 8], Razmyslov [5], Rosset [6] and, most recently, Procesi [4]. Let F be a field. For a k-tuple of matrices (A1,...,Ak) with Ai ∈ Matn(F ), let L(A1,...,Ak): Matn(F ) → Matn(F ) be the linear operator given by L(A1,...,Ak)(X) := [A1,...,Ak,X] . arXiv:2010.04679v1 [math.RA] 9 Oct 2020 Dixon and Pressman investigated the kernel of this operator in [3]. When k = 1, the kernel of L(A1) is the centralizer of A1. When k > 2n − 1, L(A1,...,Ak) is identically zero, by the Amitsur-Levitzki theorem. Conjecture 1.1 (Dixon, Pressman [3]). Suppose that 2 6 k 6 2n − 2. Then for A1,...,Ak ∈ Matn(R) in general position, the nullity d of L(A1,...,Ak) is given by (i) d = k, if k is even, 2010 Mathematics Subject Classification. 05C38, 15A24, 15A54. Key words and phrases. Amitsur-Levitsky theorem, Dixon-Pressman conjecture, directed graph, Euler- ian path, monomial order. Matthew Brassil was partially supported by a Graduate Research Fellowship from the University of British Columbia. Zinovy Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 253424-2017. 1 2 MATTHEWBRASSILANDZINOVYREICHSTEIN (ii) d = k + 1, if k is odd and n is even, and (iii) d = k + 2, if both k and n are odd. Here, as usual, R is the field of real numbers and the nullity of a linear transformation is the dimension of its kernel. Dixon and Pressman showed that d > k, k + 1 and k +2 in cases (i), (ii), and (iii), respectively, and verified computationally that equality holds for small values of n and k. Note that one may view Conjecture 1.1 and the Amitsur-Levitsky theorem as pointing in opposite directions: Conjecture 1.1 gives an upper bound on the generic nullity of L(A1,...,Ak) for 2 6 k 6 2n − 2, whereas the Amitsur-Levitsky theorem gives a lower bound for k > 2n − 1. The purpose of this paper is to prove Conjecture 1.1 in case (i). Our main result is the following. Theorem 1.2. Let k = 2r be a positive even integer and F be an infinite field whose characteristic does not divide 2(2r + 1)r!. Assume that n > r. Then for A1, A2,...,Ak ∈ Matn(F ) in general position, the nullity of L(A1, A2,...,Ak) is k. Our proof will rely on graph-theoretic techniques. To motivate it, let us briefly recall Swan’s proof of the Amitsur-Levitsky theorem. Since the standard polynomial [A1,...,Am] is multi-linear in A1,...,Am, it suffices to show that [Ea1b1 ,...,Eambm ] = 0 for any choice of a1,b1,...,am,bm ∈ {1,...,n}, as long as m > 2n. Here Eab denotes the elementary matrix with 1 in the (a,b)-position and 0s elsewhere. As we expand [Ea1b1 ,...,Eambm ], the term sgn(σ)Eaσ(1) bσ(1) ...Eaσ(m)bσ(m) contributes sgn(σ)Eaσ(1) bσ(m) to the sum if (1.1) bσ(1) = aσ(2), bσ(2) = aσ(3), ... ,bσ(m−1) = aσ(m), and 0 otherwise. Conditions (1.1) can be conveniently rephrased in graph-theoretic terms. Let G be the directed graph with n vertices, 1,...,n and m edges, e1 = (a1,b1),...,em = (am,bm). Then conditions (1.1) hold if and only if eσ(1),...,eσ(m) form an Eulerian path on G. We will say that this Eulerian path is even if σ is an even permutation and odd otherwise. This way the Amitsur-Levitsky theorem reduces to the following graph-theoretic assertion. Theorem 1.3. (Swan [7, 8]) Let G be a directed graph with n vertices and m edges. Let a and b be two of the vertices (not necessarily distinct). If m > 2n, then the number of even Eulerian paths from a to b equals the number of odd Eulerian paths from a to b. If one were to use a similar approach to prove Conjecture 1.1, one would set n (ℓ) (1.2) Aℓ = xab Eab, a,bX=1 (ℓ) 2 2 2 for ℓ =1,...,k. Here xab are kn independent variables. Each entry of the n × n matrix of L = L(A1,...,Ak) is then a multilinear polynomial of degree k in the groups of variables, (1) (k) {xab },..., {xab }. (Here we identify the linear transformation L(A1,...,Ak): Matn → Matn with its matrix in the standard basis Eab of Matn.) The coefficient of the monomial x(1) · ... · x(k) in a given position in L can again be computed as the signed sum of a1b1 akbk Eulerian paths on a certain graph. However, for k 6 2n − 2, these signed sums will no longer be identically 0. To prove Conjecture 1.1(i) in this way, one would need to assemble these coefficients into the n2 × n2 matrix L with polynomial entries, then show that the (ℓ) nullity of L over the field F (xab ) is k (or equivalently, is 6 k). We are not able to carry A GRAPH-THEORETIC APPROACH TO A CONJECTURE OF DIXON AND PRESSMAN 3 out the computations directly in this setting; the matrix L is too complicated. To prove Theorem 1.2 we will modify this approach in the following ways. (ℓ) (1) We will specialize the matrices Aℓ by setting some of the variables xab equal to 0. For the purpose of showing that null(L) 6 k, this is sufficient. In fact, we will set n2 − n entries of each Al equal to 0; the other n entries will remain independent variables. (2) We will choose A1,...,Ak so that L(A1,...,Ak) decomposes as a direct sum L(A1,...,Ak)= L0 ⊕ L1 ⊕ ... ⊕ Ln−1, where each Li is represented by an n × n matrix. This simplifies our analysis of L and reduces the problem to showing that null(L0)+null(L1)+...+null(Ln−1) 6 k. The specific matrices we will use are described in Sections 3 and 6. (3) To get a better handle on the nullities of L0,...,Ln−1, we will replace each Lj by its “matrix of initial coefficients” Ic(Lj ) with respect to a certain lexicographic monomial order on the variables xℓ,α; see Section 5. This will further simplify the computations in two ways. First, the entries of Ic(Lj ) will be integers, rather than polynomials. These integers will be obtained by counting Eulerian paths on certain graphs, as in Swan’s argument. Secondly, passing from Lj to Ic(Lj ) will allow us to focus only on the (rather special) graphs corresponding to leading monomials. We will classify these “maximal graphs” in Sections 7 and 8 and complete the proof of Theorem 1.2 in Sections 9 and 10. The last part of the proof will rely on the computations of signed counts of Eulerian paths in Section 2. The overall structure of the paper is shown in the flowchart below. Theorem 1.2 Proposition 3.2 Proposition 6.3 Proposition 7.2 Lemma 7.1 Lemma 7.3 Lemmas 2.3, 2.4 Lemma 8.1 Proposition 9.1 Lemma 8.3 Lemma 8.2 Section 10 Lemmas 2.5, 2.6 2. Preliminaries on graphs and Eulerian paths Throughout this paper our graphs will all be directed with labeled edges and vertices. An Eulerian path in a graph Γ is a path which visits every edge exactly once. We will denote by Eula(Γ) the set of Eulerian paths on Γ which begin at a vertex a. It is easy to see that any two paths in Eula(Γ) terminate at the same vertex. For an edge • → • appearing in a graph Γ we define srcΓ(e)= a and tarΓ(e)= b to be a e b the source and target vertices of the edge e respectively. We define the outdegree outdegΓ(v) to be the number of edges in Γ whose source vertex is v, and the indegree indegΓ(v) to be 4 MATTHEWBRASSILANDZINOVYREICHSTEIN the number of edges in Γ whose target vertex is v. When the graph Γ is clear from the context we will abbreviate these terms as src(e), tar(e), outdeg(v), and indeg(v). An Eulerian path beginning and ending at the same vertex is known as an Eulerian circuit. The following fundamental theorem, due to Euler, is usually stated in terms of Eulerian circuits. In the sequel we will need a variant in terms of Eulerian paths. Theorem 2.1.