SOME ALGEBRAIC IDENTITIES FOR THE α-PERMANENT HARRY CRANE Abstract. We show that the permanent of a matrix is a linear combinationofdeterminants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a moregeneral identity involving α-permanents: for arbitrary complex numbers α and β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with a discussion of the computational complexity of the α-permanent and provide some numerical illustrations. 1. Introduction The permanent of an n × n C-valued matrix M is defined by n (1) per M := Mj,σ(j), σX∈Sn Yj=1 where Sn denotes the symmetric group acting on [n] := {1,..., n}. Study of the perma- nent dates to Binet and Cauchy in the early 1800s [11]; and much of the early interest in permanents was in understanding how its resemblance of the determinant, n (2) det M := sgn(σ) Mj,σ(j), σX∈Sn Yj=1 where sgn(σ) istheparityof σ ∈ Sn, reconciled with some stark differences between the two. An early consideration, answered in the negative by Marcus and Minc [8], was whether there exists a linear transformation T such that per TM = det M for any matrix M. In his seminal paper on the #P-complete complexity class, Valiant remarked about the perplexing arXiv:1304.1772v1 [math.CO] 4 Apr 2013 relationship between the permanent and determinant, We do not know of any pair of functions, other than the permanent and determinant, for which the explicit algebraic expressions are so similar, and yet the computational complexities are apparently so different. (Valiant [17], p. 189) In this paper, we hope to give some insight to Valiant’s remark, by bringing forth the simple identity (3) (−1)n per M = (−1)↓#π det(M · π), πX∈P[n] Date: March 3, 2013. 1991 Mathematics Subject Classification. 15A15, 05E05, 05A17. Key words and phrases. permanent; α-permanent; determinant; immanant; rencontres number. 1 2 HARRY CRANE which expresses the permanent as a linear combination of determinants of block diagonal matrices. In (3), the sum is over the collection P[n] of set partitions of [n] := {1,..., n}, #π ↓j ↑j j denotes the number of blocks of π ∈ P[n], x := x(x − 1) ··· (x − j + 1) =: (−x) (−1) , and det(M · π):= b∈π det M[b], a product of determinants of the submatrices M[b] with rows and columnsQ labeled by the elements of each block of π. Equation (3) is an immediate corollary of our main identity (5) for the α-permanent [18] which, for any α ∈ C, is defined by n #σ (4) perα M := α Mj,σ(j), σX∈Sn Yj=1 where #σ denotes the number of cycles of σ ∈ Sn. The α-permanent generalizes both the permanent and the determinant: per M = per1 M and det M = per−1(−M); and, when M = Jn, the n×n matrix of all ones, (4) coincides with the generating function of the Stirling ↑n n ↑k numbers of the second kind: α = k=0 s(n, k)α , where s(n, k):= #{π ∈P[n] : #π = k}. Our main identity, P ↓#π C (5) perαβ M = β perα(M · π) for all α, β ∈ , πX∈P[n] expresses the α-permanent as a linear combination of β-permanents, for any choice of α and β. This identity, and its corollaries, could be insightful to understanding the apparent gap between the computational complexity of (1) and (2). We discuss these observations further in section 3.3, and state a conjecture about the computational complexity of the α-permanent. In addition to (5), we show other identities for the α-permanent of sums and products of matrices. We separate these identities into two main theorems, calling the first the Permanent Decomposition Theorem. Theorem 1.1 (Permanent Decomposition Theorem). For any α, β ∈ C and M ∈ Cn×n, ↓#π perαβ M = β perα(M · π), πX∈P[n] ↓j where P[n] is the collection of set partitions of [n] := {1,..., n}, β := β(β − 1) ··· (β − j + 1) and perα(M · π) = b∈π perα M[b], with M[b] denoting the submatrix of M with rows and columns labeled by the elementsQ of b ⊆ [n]. Theorem 1.2. For any α, β ∈ C and A, B ∈ Cn×n, (6) perα(A + B) = perα(AIb + BIbc ), bX⊆[n] c where Ib is a diagonal matrix with (i, i) entry 1 if i ∈ b and 0 otherwise and b is the complement of b in [n], and n = (7) perα(AB) perα(Bx) Aj,xj , xX∈[n]n Yj=1 n where Bx is the matrix whose jth row is the xjth row of B and [n] := {(i1,..., in) : 1 ≤ ij ≤ n for all j = 1 ..., n}. SOME ALGEBRAIC IDENTITIES FOR THE α-PERMANENT 3 Compared to the permanent, the α-permanent has been scarcely studied in the litera- ture. At first glance, the α-permanent may not appear as mathematically natural as the permanent or determinant: the α-permanent is not an immanant; and it is not clear what, if any, interpretation is possible for values of α other than ±1. On the other hand, the α-permanent arises naturally in statistical modeling of bosons [4, 15] (called permanental processes), as well as in connection to some well-known models in population genetics [1]. In statistical physics applications, permanental processes are complementary to deter- minantal processes, which model fermions in quantum mechanics. The Pauli exclusion principle asserts that identical fermions cannot simultaneously occupy the same quantum state, which is reflected in the property that det M = 0if tworowsof M are identical. Just as the exclusion principle does not apply to bosons, per M need not be zero if M has identical rows. Because of the practical potential of permanents, devising efficient methods (random and deterministic) for approximation is a priority. Recently, there has been some progress in this direction [5, 7, 9]; however, a provably accurate method which is also practical for large matrices is not yet available. Insection2, weproveTheorems1.1 and1.2and observeimmediate corollaries. In section 2.3, we also discuss the relationship between the α-permanent and general immanants. In section 3, we discuss computation of permanents in three contexts: in section 3.1, we discuss exact computation of the permanent for some specially structured matrices; in section 3.2, we briefly discuss some approximations of the α-permanent based on (5); in section 3.3, we use numerical approximations, see Table 1, and inspection of (5) to make a conjecture about the computational complexity of the α-permanent, which has not been studied. 2. Identities for the α-permanent In addition to (5), (6) and (7), we observe several immediate corollaries for the α- permanent. We also discuss the relationship between the α-permanent and the immanant in section 2.3. In section 2.1, we prove Theorem 1.1; in section 2.2, we prove Theorem 1.2. 2.1. Proof of Theorem 1.1. Given π ∈P[n], a set partition of [n], we define the n × n matrix (πij, 1 ≤ i, j ≤ n), also denoted π, by 1, i and j are in the same block of π, πij := ( 0, otherwise. In this way, the entries of the Hadamard product M · π coincide with the entries of M, unless a corresponding entry of π is 0. Since any π, regarded as a matrix, is the image under conjugation by σ ∈ Sn of a block diagonal matrix, and the α-permanent is invariant under conjugation by a permutation matrix, we can regard M·π as block diagonal. If we call any M · π a block diagonal projection of M, the Permanent Decomposition Theorem (Theorem 1.1) states that the αβ-permanent of a matrix is a linear combination of α-permanents of all its block diagonal projections. Also, it should be clear from (4) that perα(M · π) = perα M[b], Yb∈π where the product is over the blocks of π and M[b] denotes the sub-matrix of M with rows and columns indexed by the elements of b ⊆ [n]. Because the diagonal product 4 HARRY CRANE n j=1 Mj,σ(j)πj,σ(j) = 0 for any σ whose cycles are not a refinement of π, we have Q n #σ perα(M · π) = α Mj,σ(j), Xσ≤π Yj=1 where, for σ ∈ Sn, we write σ ≤ π to denote that each cycle of σ (as a subset of [n]) is a subset of some block of π; in other words, σ ≤ π if and only if the partition of [n] induced by σ is a refinement of π. To prove (5), we begin with the right-hand side: let α, β ∈ C and M ∈ Cn×n, then n ↓#π ↓#π #σ β perα(M · π) = β α Mj,σ(j) πX∈P[n] πX∈P[n] Xσ≤π Yj=1 n #σ ↓#π = α Mj,σ(j) β σX∈Sn Yj=1 Xπ≥σ n #σ #σ ↓j = α Mj,σ(j) s(#σ, j)β σX∈Sn Yj=1 Xj=1 n #σ = (αβ) Mj,σ(j) σX∈Sn Yj=1 = perαβ M, where s(n, k) := #{partitions of [n] with exactly k blocks} is the (n, k) Stirling number of the n n ↓k second kind, whose generating function is x := k=1 s(n, k)x .