c 2000 Society for Industrial and Applied Mathematics "

SIAM J. COMPUT. Vol. 30, No. 3, pp. 1023–1040

THE COMPUTATIONAL COMPLEXITY OF IMMANANTS∗ † ¨ PETER BURGISSER

Abstract. Permanents and determinants are special cases of immanants. The latter are polynomial matrix functions defined in terms of characters of symmetric groups and corresponding to Young diagrams. Valiant has proved that the evaluation of permanents is a complete problem in both the Turing machine model (#P-completeness) as well as in his algebraic model (VNP-completeness). We show that the evaluation of immanants corresponding to hook diagrams or rectangular diagrams of polynomially growing width is both #P-complete and VNP-complete. Key words. permanents, immanants, computational complexity, algebraic completeness AMS subject classifications. 15A15, 68Q40, 68Q15 PII. S0097539798367880

1. Introduction. The permanent per(A) of an n by n matrix A = [ai,j ] is defined by per(A) :=

n !"

ai,π(i) ,

π i=1

where the summation is over all permutations π in Sn . Note that in contrast to the determinant, each term has a positive sign. From the viewpoint of computational complexity, the determinant and permanent have, in spite of the similarity in their definitions, very little in common. While there are efficient polynomial time algorithms for the evaluation of the determinant, the best-known algorithm for the evaluation of the permanent of an n by n matrix needs O(n2n ) arithmetic operations (Ryser [20]). A hypothesis due to Valiant in fact claims that the permanent cannot be computed with a polynomial number of arithmetic operations. This hypothesis is supported by Valiant’s famous result [23] stating that the problem to evaluate the permanent of a matrix with entries in {0, 1} is #Pcomplete, as well as his analogous VNP-completeness result [22] in a framework of algebraic computations. Both permanents and determinants are special cases of immanants introduced by Littlewood [16]. To define these polynomial matrix functions, we have to rely on some basic facts about the characters of the symmetric groups, which can be found for instance in the books by Boerner [2], James and Kerber [13], or Fulton and Harris [9]. It is known that the irreducible characters of the symmetric group Sn can be labeled by partitions of n, i.e., by decreasing sequences λ = (λ1 , . . . , λs ) of natural numbers adding up to n. A partition will be identified with its (Young) diagram {(i, j) | 1 ≤ j ≤ λi }, which can be visualized as a left-justified # arrangement of λi boxes in the ith row. (Compare Figure 3.1.) We call |λ| := i λi the size and λ1 the width of λ and will use the notation λ " n to express that λ is a partition or a diagram of size n. A diagram is called rectangular iff λ1 = · · · = λs . ∗ Received by the editors July 29, 1998; accepted for publication (in revised form) April 20, 1999; published electronically August 24, 2000. The results of this work were announced in Proceedings of the 10th Conference on Formal Power Series and Algebraic Combinatorics, University of Toronto, Canada, 1998, pp. 115–126. http://www.siam.org/journals/sicomp/30-3/36788.html † Department of Mathematics and Computer Science, University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany ([email protected]).

1023

1024

¨ PETER BURGISSER

Let χλ : Sn → Z denote the irreducible character of the symmetric group Sn corresponding to the diagram λ " n. The immanant of an n by n matrix A = [ai,j ] corresponding to λ is defined by imλ (A) :=

!

π∈Sn

χλ (π)

n "

ai,π(i) .

i=1

For the “horizontal” diagrams given by λ = (n) we have χλ = 1, and the corresponding immanants specialize to the permanents. In the case of the “vertical” diagrams given by λ = (1, . . . , 1) we get χλ = sgn, and the immanants specialize to the determinants. These diagrams are special cases of those described by (k, 1, . . . , 1), which are called hook diagrams because of their shape. By hook immanants or rectangular immanants we will understand the immanant polynomials corresponding to diagrams of the corresponding shape. The reader may find some algebraic properties of immanants in Merris [17, 18]. Our interest for immanants stems from the fact that they constitute a natural parameterized set of polynomials, which allows us to study the change of computational complexity from easy (polynomial time computable) to difficult (complete) as the diagram λ of size n varies between the vertical and the horizontal diagram. To make this more specific, think for instance of the set of hook diagrams (k, 1, . . . , 1) " n as the parameter k varies between 1 and n. In the previous paper [3] we have developed a fast algorithm to evaluate representations of general linear groups. As a byproduct, we obtained an algorithm to evaluate the immanant imλ at a matrix A with nonscalar cost proportional to n2 sλ dλ , where sλ and dλ denote the number of standard tableaus and semistandard tableaus on the diagram of λ, respectively. This upper bound improves previous bounds due to Hartmann [11] and Barvinok [1]. In the present article we complement the upper bounds in [3] by completeness results for certain families of immanants. The results will be formulated in Valiant’s algebraic P-NP theory, centering around the notion of VNP-completeness. The main features of this theory are recalled in section 2. Note that all VNP-completeness statements in this article refer to a ground field of characteristic zero. A sequence (λ(n) ) of diagrams such that the size of λ(n) is polynomially bounded in n will be called a p-sequence of diagrams. If, additionally, the width of λ(n) is growing at least polynomially in the size of λ(n) , then we call such a sequence to be of polynomially growing width. The corresponding families of immanant polynomials will also be called so. We have the following conjecture. Conjecture 1.1. Any family of immanant polynomials of polynomially growing width is VNP-complete. The achievement of this paper is the proof of this conjecture in two special situations: for hook and rectangular immanants. The statement for hook immanants generalizes a result by Hartmann [11], while the claim for rectangular immanants answers an open problem posed by Strassen [21, Problem 14.2]. Theorem 1.2. Any family of hook immanants or rectangular immanants of polynomially growing width is VNP-complete. For sequences of diagrams of bounded width we have so far no clear idea of the complexity of the corresponding immanants. We raise the following question. Problem 1.3. Is the family of rectangular immanants corresponding to rectangles of width 2 VNP-complete?

THE COMPUTATIONAL COMPLEXITY OF IMMANANTS

1025

We remark that families of hook immanants of bounded width are p-computable. This is an immediate consequence of the upper complexity bound in [3] mentioned before. Our proofs also yield #P-completeness results for the problem to evaluate immanants at matrices A with entries in {0, 1}. However, note that imχ (A) may be negative, with absolute value bounded by n! nn ≤ n2n for A ∈ {0, 1}n×n . We will therefore interpret the evaluation problem below as the modified problem to compute imχ (A) + n2n from A ∈ {0, 1}n×n . Corollary 1.4. Assume in Theorem 1.2 that the sequence of diagrams is polynomial time computable. Then the problem to evaluate the corresponding immanant at a given matrix with entries in {0, 1} is #P-complete. The paper is organized as follows. In section 2 we recall the main features of Valiant’s algebraic P-NP theory. The goal of section 3 is the proof of an auxiliary result (Lemma 3.1) which is crucial for our completeness proofs. It expresses values of characters corresponding to rectangular diagrams by characters of hook diagrams. Section 4 is devoted to the proof that families of immanants are indeed p-definable. In section 5, we provide the proof of Theorem 1.2 in several steps. The general strategy is to identify alternating sums of immanants corresponding to smaller partitions as a projection of a given immanant (Lemma 5.1), by applying the Murnaghan– Nakayama rule for the characters of the symmetric group. In combination with Lemma 3.1 we exhibit alternating sums of hook immanants as projections of rectangular immanants. This is then combined with technical results, which allows to obtain permanents or Hamilton cycle polynomials as projections of linear combinations of hook immanants. 2. Valiant’s algebraic model of NP-completeness. We briefly recall the main features of Valiant’s algebraic P-NP theory. For detailed expositions we refer to the survey by von zur Gathen [10] and the books by B¨ urgisser, Clausen, and Shokrollahi [7, Chapter 21] and B¨ urgisser [6]. We will adopt the following useful convention: we denote matrix functions with small letters, but the corresponding function evaluated at a matrix with distinct indeterminate entries is written in capitals. For instance, per(A) is the permanent of the n by n matrix A, and PERn =

n ! "

Xi,π(i)

π∈Sn i=1

denotes the permanent of an n by n matrix with indeterminate entries Xi,j . Likewise, IMλ denotes the immanant polynomial corresponding to the diagram λ " n. Throughout the paper, the discussion will refer to a fixed field k of characteristic zero. (The reader may assume k = Q.) By a p-family we understand a sequence (fn ) of multivariate polynomials fn ∈ k[X1 , . . . , Xv(n) ] such that the number of variables v(n) as well as the degree deg fn are p-bounded functions of n, i.e., these functions are majorized by a polynomial in n. Interesting examples are the permanent family PER = (PERn ), the determinant family DET = (DETn ), and the family HC = (HCn ) of Hamilton cycle polynomials defined by HCn =

n !"

π i=1

Xi,π(i) ,

¨ PETER BURGISSER

1026

where the sum is over all cycles π ∈ Sn of length n. Note that the value of HCn at the adjacency matrix of a digraph equals the number of its Hamilton cycles. Let L(fn ) denote the total complexity of fn ∈ k[X1 , . . . , Xv(n) ], that is, the minimum number of arithmetic operations +, −, ∗ to compute fn from the variables Xi and constants in k by a straight-line program. We call a p-family p-computable iff L(fn ) is p-bounded in n. The p-computable families constitute the complexity class VP. A p-family (fn ) is called p-definable iff there exists a p-computable family (gn ), gn ∈ k[X1 , . . . , Xu(n) ], such that for all n fn (X1 , . . . , Xv(n) ) =

!

gn (X1 , . . . , Xv(n) , ev(n)+1 , . . . , eu(n) ).

e∈{0,1}u(n)−v(n)

The set of p-definable families form the complexity class VNP. We will employ a very simple notion of reduction. A polynomial fn is called a projection of a polynomial gm ∈ k[X1 , . . . , Xu ], written fn ≤ gm , iff fn (X1 , . . . , Xv(n) ) = gm (a1 , . . . , au ) for some ai ∈ k ∪ {X1 , . . . , Xv(n) }. That is, fn can be derived from gm through substitution by indeterminates and constants. A p-family (fn ) is called a p-projection of a family (gm ) iff there is a p-bounded function t : N → N such that fn is a projection of gt(n) for all n. Finally, a p-definable family (gm ) is called VNP-complete iff any (fn ) ∈ VNP is a p-projection of (gm ). In [22] Valiant proved that the p-families PER of permanents and HC of Hamilton cycles polynomials are VNP-complete (over fields k of characteristic different from two, which is a general assumption in this paper). Thus PER is not p-computable iff Valiant’s hypothesis VP (= VNP is true. One can prove that the “generating functions” corresponding to several NPcomplete graph problems like cliques, graph factors, Hamilton cycles in planar graphs, etc. yield VNP-complete families as well (see [6, Chapter 3]). In fact, Valiant’s hypothesis can be considered as a nonuniform algebraic counterpart of the well-known hypothesis P (= NP due to Cook [8]. For results relating these two hypotheses, see [4, 6]. We mention in passing that, by contrast with the classical P-NP theory, one knows specific p-definable families over finite fields, which are neither VNP-complete, nor p-computable, provided the polynomial hierarchy does not collapse (cf. [5, 6]). For later use, we state some results going back to Valiant [24]; detailed proofs can be found in [6]. The first result shows that the complexity class VNP is closed under various natural operations. Proposition 2.1. Let (fn ) and (gn ) be p-definable; say fn ∈ k[X1 , . . . , Xv(n) ]. Then the following hold. (i) Sum and product. (fn + gn ) and (fn · gn ) are p-definable. (ii) Substitution. (fn (g1 , . . . , gv(n) )) is p-definable. (iii) Coefficient. If hn ∈ k[Xu(n)+1 , . . . , Xv(n) ] is the coefficient of some power i

u(n) product X1i1 · · · Xu(n) in fn for some u(n) ≤ v(n), then the family (hn ) is p-definable. The second result is a useful criterion for p-definability which connects the nonuniform counting complexity class #P/poly to the class VNP. Note that functions ϕ, which are computable in polynomial time on a Turing machine, are clearly contained in the class #P/poly. (For the definition of #P see [23]; a general definition of nonuniform complexity classes like #P/poly can be found in Karp and Lipton [15].)

THE COMPUTATIONAL COMPLEXITY OF IMMANANTS

1027

Proposition 2.2. Suppose ϕ : {0, 1}∗ → N is a function in the class #P/poly. Then the family (fn ) of polynomials defined by ! ϕ(e)X1e1 · · · Xnen fn = e∈{0,1}n

is p-definable. 3. Character formulas for the symmetric group. We first recall some character formulas for the symmetric group for later use. These formulas are then applied to derive Lemma 3.1, which is a crucial ingredient of our completeness proofs. More information on the characters of the symmetric groups can be found in the books by Boerner [2], James and Kerber [13], or Fulton and Harris [9]. We recall that λ " n means that λ = (λ1 , . . . , λs ) is a partition of n. The irreducible character of Sn corresponding to λ is denoted by χλ . To a partition λ we may assign the strictly decreasing sequence % = [%1 , . . . , %s ] := (λ1 , . . . , λs ) + (s − 1, s − 2, . . . , 1, 0) $% # %i = n + 2s . (We use square brackets to distinguish % notain Ns which satisfies tionally from a partition λ.) It is useful to index the irreducible characters of Sn by s such sequences, thus we$ set % χ$ := χλ . We can extend this definition to any % ∈ N # s satisfying %i = n + 2 by requiring the function % )→ χ$ to be alternating. In particular, χ$ vanishes if two components of % are equal. We also include the case n = 0 by setting χ[s−1,...,0] (1) := 1. Conjugacy classes of permutations in Sn are described by their # cycle format (ρ1 , . . . , ρn ), where ρi denotes the number of i-cycles. Clearly, i iρi = n. It will be convenient to write cycle formats in frequency notation ρ = 1ρ1 · · · nρn , or shorter ρ |= n in order to express that ρ is a cycle format of n. Moreover, we set χ$ (ρ) := χ$ (π), where π is any permutation with cycle format ρ. Let us,i := Z1i + Z2i + · · · + Zsi denote the ith elementary power sum in the indeterminates Z1 , . . . , Zs , and let " ∆s := det(Zis−j )1≤i,j≤s = (Zi − Zj ) i

SIAM J. COMPUT. Vol. 30, No. 3, pp. 1023–1040

THE COMPUTATIONAL COMPLEXITY OF IMMANANTS∗ † ¨ PETER BURGISSER

Abstract. Permanents and determinants are special cases of immanants. The latter are polynomial matrix functions defined in terms of characters of symmetric groups and corresponding to Young diagrams. Valiant has proved that the evaluation of permanents is a complete problem in both the Turing machine model (#P-completeness) as well as in his algebraic model (VNP-completeness). We show that the evaluation of immanants corresponding to hook diagrams or rectangular diagrams of polynomially growing width is both #P-complete and VNP-complete. Key words. permanents, immanants, computational complexity, algebraic completeness AMS subject classifications. 15A15, 68Q40, 68Q15 PII. S0097539798367880

1. Introduction. The permanent per(A) of an n by n matrix A = [ai,j ] is defined by per(A) :=

n !"

ai,π(i) ,

π i=1

where the summation is over all permutations π in Sn . Note that in contrast to the determinant, each term has a positive sign. From the viewpoint of computational complexity, the determinant and permanent have, in spite of the similarity in their definitions, very little in common. While there are efficient polynomial time algorithms for the evaluation of the determinant, the best-known algorithm for the evaluation of the permanent of an n by n matrix needs O(n2n ) arithmetic operations (Ryser [20]). A hypothesis due to Valiant in fact claims that the permanent cannot be computed with a polynomial number of arithmetic operations. This hypothesis is supported by Valiant’s famous result [23] stating that the problem to evaluate the permanent of a matrix with entries in {0, 1} is #Pcomplete, as well as his analogous VNP-completeness result [22] in a framework of algebraic computations. Both permanents and determinants are special cases of immanants introduced by Littlewood [16]. To define these polynomial matrix functions, we have to rely on some basic facts about the characters of the symmetric groups, which can be found for instance in the books by Boerner [2], James and Kerber [13], or Fulton and Harris [9]. It is known that the irreducible characters of the symmetric group Sn can be labeled by partitions of n, i.e., by decreasing sequences λ = (λ1 , . . . , λs ) of natural numbers adding up to n. A partition will be identified with its (Young) diagram {(i, j) | 1 ≤ j ≤ λi }, which can be visualized as a left-justified # arrangement of λi boxes in the ith row. (Compare Figure 3.1.) We call |λ| := i λi the size and λ1 the width of λ and will use the notation λ " n to express that λ is a partition or a diagram of size n. A diagram is called rectangular iff λ1 = · · · = λs . ∗ Received by the editors July 29, 1998; accepted for publication (in revised form) April 20, 1999; published electronically August 24, 2000. The results of this work were announced in Proceedings of the 10th Conference on Formal Power Series and Algebraic Combinatorics, University of Toronto, Canada, 1998, pp. 115–126. http://www.siam.org/journals/sicomp/30-3/36788.html † Department of Mathematics and Computer Science, University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany ([email protected]).

1023

1024

¨ PETER BURGISSER

Let χλ : Sn → Z denote the irreducible character of the symmetric group Sn corresponding to the diagram λ " n. The immanant of an n by n matrix A = [ai,j ] corresponding to λ is defined by imλ (A) :=

!

π∈Sn

χλ (π)

n "

ai,π(i) .

i=1

For the “horizontal” diagrams given by λ = (n) we have χλ = 1, and the corresponding immanants specialize to the permanents. In the case of the “vertical” diagrams given by λ = (1, . . . , 1) we get χλ = sgn, and the immanants specialize to the determinants. These diagrams are special cases of those described by (k, 1, . . . , 1), which are called hook diagrams because of their shape. By hook immanants or rectangular immanants we will understand the immanant polynomials corresponding to diagrams of the corresponding shape. The reader may find some algebraic properties of immanants in Merris [17, 18]. Our interest for immanants stems from the fact that they constitute a natural parameterized set of polynomials, which allows us to study the change of computational complexity from easy (polynomial time computable) to difficult (complete) as the diagram λ of size n varies between the vertical and the horizontal diagram. To make this more specific, think for instance of the set of hook diagrams (k, 1, . . . , 1) " n as the parameter k varies between 1 and n. In the previous paper [3] we have developed a fast algorithm to evaluate representations of general linear groups. As a byproduct, we obtained an algorithm to evaluate the immanant imλ at a matrix A with nonscalar cost proportional to n2 sλ dλ , where sλ and dλ denote the number of standard tableaus and semistandard tableaus on the diagram of λ, respectively. This upper bound improves previous bounds due to Hartmann [11] and Barvinok [1]. In the present article we complement the upper bounds in [3] by completeness results for certain families of immanants. The results will be formulated in Valiant’s algebraic P-NP theory, centering around the notion of VNP-completeness. The main features of this theory are recalled in section 2. Note that all VNP-completeness statements in this article refer to a ground field of characteristic zero. A sequence (λ(n) ) of diagrams such that the size of λ(n) is polynomially bounded in n will be called a p-sequence of diagrams. If, additionally, the width of λ(n) is growing at least polynomially in the size of λ(n) , then we call such a sequence to be of polynomially growing width. The corresponding families of immanant polynomials will also be called so. We have the following conjecture. Conjecture 1.1. Any family of immanant polynomials of polynomially growing width is VNP-complete. The achievement of this paper is the proof of this conjecture in two special situations: for hook and rectangular immanants. The statement for hook immanants generalizes a result by Hartmann [11], while the claim for rectangular immanants answers an open problem posed by Strassen [21, Problem 14.2]. Theorem 1.2. Any family of hook immanants or rectangular immanants of polynomially growing width is VNP-complete. For sequences of diagrams of bounded width we have so far no clear idea of the complexity of the corresponding immanants. We raise the following question. Problem 1.3. Is the family of rectangular immanants corresponding to rectangles of width 2 VNP-complete?

THE COMPUTATIONAL COMPLEXITY OF IMMANANTS

1025

We remark that families of hook immanants of bounded width are p-computable. This is an immediate consequence of the upper complexity bound in [3] mentioned before. Our proofs also yield #P-completeness results for the problem to evaluate immanants at matrices A with entries in {0, 1}. However, note that imχ (A) may be negative, with absolute value bounded by n! nn ≤ n2n for A ∈ {0, 1}n×n . We will therefore interpret the evaluation problem below as the modified problem to compute imχ (A) + n2n from A ∈ {0, 1}n×n . Corollary 1.4. Assume in Theorem 1.2 that the sequence of diagrams is polynomial time computable. Then the problem to evaluate the corresponding immanant at a given matrix with entries in {0, 1} is #P-complete. The paper is organized as follows. In section 2 we recall the main features of Valiant’s algebraic P-NP theory. The goal of section 3 is the proof of an auxiliary result (Lemma 3.1) which is crucial for our completeness proofs. It expresses values of characters corresponding to rectangular diagrams by characters of hook diagrams. Section 4 is devoted to the proof that families of immanants are indeed p-definable. In section 5, we provide the proof of Theorem 1.2 in several steps. The general strategy is to identify alternating sums of immanants corresponding to smaller partitions as a projection of a given immanant (Lemma 5.1), by applying the Murnaghan– Nakayama rule for the characters of the symmetric group. In combination with Lemma 3.1 we exhibit alternating sums of hook immanants as projections of rectangular immanants. This is then combined with technical results, which allows to obtain permanents or Hamilton cycle polynomials as projections of linear combinations of hook immanants. 2. Valiant’s algebraic model of NP-completeness. We briefly recall the main features of Valiant’s algebraic P-NP theory. For detailed expositions we refer to the survey by von zur Gathen [10] and the books by B¨ urgisser, Clausen, and Shokrollahi [7, Chapter 21] and B¨ urgisser [6]. We will adopt the following useful convention: we denote matrix functions with small letters, but the corresponding function evaluated at a matrix with distinct indeterminate entries is written in capitals. For instance, per(A) is the permanent of the n by n matrix A, and PERn =

n ! "

Xi,π(i)

π∈Sn i=1

denotes the permanent of an n by n matrix with indeterminate entries Xi,j . Likewise, IMλ denotes the immanant polynomial corresponding to the diagram λ " n. Throughout the paper, the discussion will refer to a fixed field k of characteristic zero. (The reader may assume k = Q.) By a p-family we understand a sequence (fn ) of multivariate polynomials fn ∈ k[X1 , . . . , Xv(n) ] such that the number of variables v(n) as well as the degree deg fn are p-bounded functions of n, i.e., these functions are majorized by a polynomial in n. Interesting examples are the permanent family PER = (PERn ), the determinant family DET = (DETn ), and the family HC = (HCn ) of Hamilton cycle polynomials defined by HCn =

n !"

π i=1

Xi,π(i) ,

¨ PETER BURGISSER

1026

where the sum is over all cycles π ∈ Sn of length n. Note that the value of HCn at the adjacency matrix of a digraph equals the number of its Hamilton cycles. Let L(fn ) denote the total complexity of fn ∈ k[X1 , . . . , Xv(n) ], that is, the minimum number of arithmetic operations +, −, ∗ to compute fn from the variables Xi and constants in k by a straight-line program. We call a p-family p-computable iff L(fn ) is p-bounded in n. The p-computable families constitute the complexity class VP. A p-family (fn ) is called p-definable iff there exists a p-computable family (gn ), gn ∈ k[X1 , . . . , Xu(n) ], such that for all n fn (X1 , . . . , Xv(n) ) =

!

gn (X1 , . . . , Xv(n) , ev(n)+1 , . . . , eu(n) ).

e∈{0,1}u(n)−v(n)

The set of p-definable families form the complexity class VNP. We will employ a very simple notion of reduction. A polynomial fn is called a projection of a polynomial gm ∈ k[X1 , . . . , Xu ], written fn ≤ gm , iff fn (X1 , . . . , Xv(n) ) = gm (a1 , . . . , au ) for some ai ∈ k ∪ {X1 , . . . , Xv(n) }. That is, fn can be derived from gm through substitution by indeterminates and constants. A p-family (fn ) is called a p-projection of a family (gm ) iff there is a p-bounded function t : N → N such that fn is a projection of gt(n) for all n. Finally, a p-definable family (gm ) is called VNP-complete iff any (fn ) ∈ VNP is a p-projection of (gm ). In [22] Valiant proved that the p-families PER of permanents and HC of Hamilton cycles polynomials are VNP-complete (over fields k of characteristic different from two, which is a general assumption in this paper). Thus PER is not p-computable iff Valiant’s hypothesis VP (= VNP is true. One can prove that the “generating functions” corresponding to several NPcomplete graph problems like cliques, graph factors, Hamilton cycles in planar graphs, etc. yield VNP-complete families as well (see [6, Chapter 3]). In fact, Valiant’s hypothesis can be considered as a nonuniform algebraic counterpart of the well-known hypothesis P (= NP due to Cook [8]. For results relating these two hypotheses, see [4, 6]. We mention in passing that, by contrast with the classical P-NP theory, one knows specific p-definable families over finite fields, which are neither VNP-complete, nor p-computable, provided the polynomial hierarchy does not collapse (cf. [5, 6]). For later use, we state some results going back to Valiant [24]; detailed proofs can be found in [6]. The first result shows that the complexity class VNP is closed under various natural operations. Proposition 2.1. Let (fn ) and (gn ) be p-definable; say fn ∈ k[X1 , . . . , Xv(n) ]. Then the following hold. (i) Sum and product. (fn + gn ) and (fn · gn ) are p-definable. (ii) Substitution. (fn (g1 , . . . , gv(n) )) is p-definable. (iii) Coefficient. If hn ∈ k[Xu(n)+1 , . . . , Xv(n) ] is the coefficient of some power i

u(n) product X1i1 · · · Xu(n) in fn for some u(n) ≤ v(n), then the family (hn ) is p-definable. The second result is a useful criterion for p-definability which connects the nonuniform counting complexity class #P/poly to the class VNP. Note that functions ϕ, which are computable in polynomial time on a Turing machine, are clearly contained in the class #P/poly. (For the definition of #P see [23]; a general definition of nonuniform complexity classes like #P/poly can be found in Karp and Lipton [15].)

THE COMPUTATIONAL COMPLEXITY OF IMMANANTS

1027

Proposition 2.2. Suppose ϕ : {0, 1}∗ → N is a function in the class #P/poly. Then the family (fn ) of polynomials defined by ! ϕ(e)X1e1 · · · Xnen fn = e∈{0,1}n

is p-definable. 3. Character formulas for the symmetric group. We first recall some character formulas for the symmetric group for later use. These formulas are then applied to derive Lemma 3.1, which is a crucial ingredient of our completeness proofs. More information on the characters of the symmetric groups can be found in the books by Boerner [2], James and Kerber [13], or Fulton and Harris [9]. We recall that λ " n means that λ = (λ1 , . . . , λs ) is a partition of n. The irreducible character of Sn corresponding to λ is denoted by χλ . To a partition λ we may assign the strictly decreasing sequence % = [%1 , . . . , %s ] := (λ1 , . . . , λs ) + (s − 1, s − 2, . . . , 1, 0) $% # %i = n + 2s . (We use square brackets to distinguish % notain Ns which satisfies tionally from a partition λ.) It is useful to index the irreducible characters of Sn by s such sequences, thus we$ set % χ$ := χλ . We can extend this definition to any % ∈ N # s satisfying %i = n + 2 by requiring the function % )→ χ$ to be alternating. In particular, χ$ vanishes if two components of % are equal. We also include the case n = 0 by setting χ[s−1,...,0] (1) := 1. Conjugacy classes of permutations in Sn are described by their # cycle format (ρ1 , . . . , ρn ), where ρi denotes the number of i-cycles. Clearly, i iρi = n. It will be convenient to write cycle formats in frequency notation ρ = 1ρ1 · · · nρn , or shorter ρ |= n in order to express that ρ is a cycle format of n. Moreover, we set χ$ (ρ) := χ$ (π), where π is any permutation with cycle format ρ. Let us,i := Z1i + Z2i + · · · + Zsi denote the ith elementary power sum in the indeterminates Z1 , . . . , Zs , and let " ∆s := det(Zis−j )1≤i,j≤s = (Zi − Zj ) i