HAL Id: hal-02158533
https://hal.archives-ouvertes.fr/hal-02158533
Submitted on 25 Jul 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Permanent V. Determinant: An Exponential Lower
Bound Assuming Symmetry
Joseph Landsberg, Nicolas Ressayre
To cite this version:
Permanent v. determinant: an exponential lower bound
assuming symmetry
[Extended Abstract]
J.M. Landsberg
Institute for Clarity in Documentation 1932 Wallamaloo Lane Wallamaloo, New Zealand
[email protected]
Nicolas Ressayre
Institute for Clarity in Documentation P.O. Box 1212
Dublin, Ohio 43017-6221
[email protected]
ABSTRACT
Grenet’s determinantal representation for the permanent is optimal among determinantal representations that are equiv-ariant with respect to left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal rep-resentation of the permanent must be polynomially related to one with such symmetry, then Valiant’s conjecture on permanent v. determinant is true.
Categories and Subject Descriptors
H.4 [Information Systems Applications]: Miscellaneous
General Terms
Theory
Keywords
Geometric Complexity Theory, determinant, permanent, MSC 68Q15 (20G05)
1.
INTRODUCTION
The determinant detn(x) := X σ∈Sn sgn(σ)x1σ(1)x 2 σ(2)· · · x n σ(n), (1) is a homogeneous polynomial of degree n on the space of n × n matrices Matn(C). Here Sn denotes the group of permutations on n elements and sgn(σ) denotes the sign of the permutation σ.Despite its formula with n! terms, detn can be evaluated quickly, e.g., using Gaussian elimination, which exploits the large symmetry group of the determinant, e.g., detn(x) = ∗A full version of this paper is available as [14].
detn(AxB−1) for any n × n matrices A, B with determinant equal to one.
We will work exclusively over the complex numbers and with homogeneous polynomials, the latter restriction only for convenience. L. Valiant showed that given a homoge-neous polynomial P (y) in M variables, there exists an n and an affine linear map
˜
A : CM → Matn(C) y 7→ Λ + A(y)
where Λ is a constant matrix and y 7→ A(y) is a linear map CM → Matn(C) such that P = detn◦ ˜A. Such ˜A is called a determinantal representation of P . When M = m2and P is the permanent polynomial
permm(y) := X σ∈Sm yσ(1)y1 2σ(2)· · · y m σ(m), (2) L. Valiant showed that one can take n = O(2m). As an alge-braic analog of the P 6= NP conjecture, he also conjectured that one cannot do much better:
Conjecture 1 (Valiant [21]). Let n(m) be a func-tion of m such that there exist affine linear maps
˜
Am: Cm2→ Matn(m)(C) satisfying
permm= detn(m)◦ ˜Am. (3) Then n(m) grows faster than any polynomial.
To measure progress towards Conjecture 1, define dc(permm) to be the smallest n(m) such that there exists ˜Am satis-fying (3). The conjecture is that dc(permm) grows faster than any polynomial in m. Lower bounds on dc(permm) are: dc(permm) > m (Marcus and Minc [15]), dc(permm) > 1.06m (Von zur Gathen [23]), dc(permm) >√2m − O(√m) (Meshulam, reported in [23], and Cai [4]), with the cur-rent world record dc(permm) ≥ m
2
2 [16] by Mignon and the second author. (Over R, Yabe recently showed that dcR(permm) ≥ m2− 2m + 2 [24], and in [5] Cai, Chen and Li extended the m2
2 bound to arbitrary fields.)
GLM denote the group of invertible linear maps CM → CM. For P ∈ SmCM ∗, a homogeneous polynomial of degree m on CM, let
GP := {g ∈ GLM | P (g−1y) = P (y) ∀y ∈ CM} denote the symmetry group of P . For example Gdetn ' (SLn× SLn)/C∗
o Z2 [9], where the SLn× SLn invariance comes from det(AxB−1) = (det A det B−1) det(x) and the Z2is because detn(x) = detn(xT) where xT is the transpose of the matrix x.
As observed in [18], the permanent (resp. determinant) is characterized by its symmetries in the sense that any poly-nomial P ∈ SmCm
2∗
with a symmetry group GP such that GP ⊇ Gpermm (resp. GP ⊇ Gdetm) is a scalar multiple of the permanent (resp. determinant). This property is the cornerstone of GCT. The program outlined in [18, 19] is an approach to Valiant’s conjecture based on the functions on GLn2 that respect the symmetry group Gdetn, i.e., are invariant under the action of Gdetn.
Guided by the principles of GCT, we ask:
What are the ˜A that respect the symmetry group of the permanent?
To make this question precise:
Definition 1. Let ˜A : CM
−→ Matn(C) be a determinan-tal representation of P ∈ Sm
CM ∗. Define GA= {g ∈ Gdetn| g · Λ = Λ and g · A(C
M
) = A(CM)}, the symmetry group of the determinantal representation ˜A of P .
The group GA comes with a representation ρA : GA −→ GL(A(CM)) obtained by restricting the action to A(CM). We assume that P cannot be expressed using M − 1 vari-ables, i.e., that P 6∈ SmCM −1 for any hyperplane CM −1 ⊂ CM ∗. Then A : CM −→ A(CM) is bijective. Let A−1 : A(CM
) −→ CM denote its inverse. Set ¯
ρA : GA−→ GLM (4) g 7−→ A ◦ ρA(g) ◦ A−1.
Then ρA(GA) ⊆ GP.
Definition 2. We say ˜A is an equivariant representation of P if (4) surjects onto GP.
If G is a subgroup of GP, we say that ˜A is G-equivariant if G is contained in the image of ¯ρA.
Example 1. Let Q =PMj=1z2j ∈ S2CM ∗ be a nondegen-erate quadric. Then GQ = O(M ) where O(M ) = {B ∈ GLM | B−1
= BT} is the orthogonal group, as for such B,
B · Q = P
i,j,kBi,jBk,jzizk = P
ijδijzizj = Q. Consider the determinantal representation
Q = detM +1 0 −z1 · · · −zM z1 1 . . . . .. zM 1 . (5)
For (λ, B) ∈ GQ, define an action on Z ∈ MatM +1(C) by Z 7→λ 0 0 B Zλ −1 0 0 B −1 . Write X = x1 . . . xM so ˜ A = 0 −X T X IdM .
The relation B−1= BT implies λ 0 0 B · 0 −X T X IdM ·λ −1 0 0 B −1 = 0 −(λBX)T λBX IdM . Taking detM +1 on both sides gives
λ2Q(X) = (λ, B) · Q(X). Thus ˜A is an equivariant representation of Q.
Definition 3. For P ∈ SmCM ∗, define the equivariant determinantal complexity of P , denoted edc(P ), to be the smallest n such that there is an equivariant determinantal representation of P .
Of course edc(P ) ≥ dc(P ). We do not know if edc(P ) is finite in general. Our main result is that edc(permm) is exponential in m.
2.
RESULTS
2.1
Main Theorem
Theorem 1. Let m ≥ 3. Then edc(permm) = 2mm −1 ∼ 4m.
There are several instances in complexity theory where an optimal algorithm partially respects symmetry, e.g. Strassen’s algorithm for 2 × 2 matrix multiplication respects the Z3-symmetry of the matrix multiplication operator (see [13, §4.2]), but not the GL×32 symmetry.
For the purposes of Valiant’s conjecture, we ask the weaker question:
Question 1. Does there exist a polynomial e(d) such that edc(permm) ≤ e(dc(permm))?
Theorem 1 implies:
We have no opinion as to what the answer to Question 1 should be, but as it provides a new potential path to proving Valiant’s conjecture, it merits further investigation. Note that Question 1 is a flip in the terminology of [17], since a positive answer is an existence result. It fits into the more general question: When an object has symmetry, does it ad-mit an optimal expression that preserves its symmetry?
Example 2. Let T ∈ W⊗dbe a symmetric tensor, i.e. T ∈ SdW ⊂ W⊗d. Say T can be written as a sum of r rank one tensors, then P. Comon conjectures [6] that it can be written as a sum of r rank one symmetric tensors.
Example 3. The optimal Waring decomposition of x1· · · xn, dating back at least to [8] and proved to be optimal in [20] is x1· · · xn= 1 2n−1n! X ∈{−1,1}n 1=1 n X j=1 jxjnΠni=1i, (6)
a sum with 2n−1 terms. This decomposition has an Sn−1-symmetry but not an Sn-Sn−1-symmetry, nor does it preserve the action of the torus TSLn of diagonal matrices with determi-nant one. One can obtain an Sn-invariant expression by doubling the size:
x1· · · xn= 1 2nn! X ∈{−1,1}n n X j=1 jxjnΠni=1i, , (7) because (−x1+ 2x2+ · · · + nxn)n(−1)2· · · n =(−1)n(x1+ (−2)x2+ · · · + (−n)xn)n(−1)2· · · n =(x1+ (−2)x2+ · · · + (−n)xn)n(−2) · · · (−n).
The optimal Waring decomposition of the permanent is not known, but it is known to be of size greater than bn/2cn 2∼ 4n/√n. The Ryser-Glynn formula [10] is
permn(x) = 2−n+1 X ∈{−1,1}n 1=1 Y 1≤i≤n X 1≤j≤n ijxi,j, (8)
the outer sum taken over n-tuples (1= 1, 2, · · · , n). This Sn−1× Sn−1-invariant formula can also be made Sn× Sn-invariant by enlarging it by a factor of 4, to get a Sn× Sn homogeneous depth three formula that is within a factor of four of the best known. Then expanding each monomial above, using Equation (7), one gets a Sn× Sn-Waring ex-pression within a factor of O(√n) of the lower bound.
Example 4. Examples regarding equivariant representa-tions of SN-invariant funcrepresenta-tions from the Boolean world give inconclusive indications regarding Question 1.
The M ODm-degree of a Boolean function f (x1, . . . , xN) is the smallest degree of any polynomial P ∈ Z[x1, . . . , xN] such that f (x) = 0 if and only if P (x) = 0 for all x ∈ {0, 1}N. The known upper bound for the M ODm-degree of the Boolean OR function (OR(x1, . . . , xN) = 1 if any xj = 1 and is zero if all xj = 0) is attained by symmetric
polynomials [3]. Moreover in [3] it is also shown that this bound cannot be improved with symmetric polynomials, and it is far from the known lower bound.
The boolean majority function M AJ (x1, . . . , xN) takes on 1 if at least half the xj = 1 and zero otherwise. The best monotone Boolean formula for M AJ [22] is polynomial in N and attained using random functions, and it is expected that the only symmetric monotone formula for majority is the trivial one, disjunction of all n2-size subsets (or its dual), which is of exponential size.
Question 2. Does every P that is determined by its sym-metry group admit an equivariant determinantal representa-tion? For those P that do, how much larger must such a determinantal representation be from the size of a minimal one?
2.2
Grenet’s formulas
The starting point of our investigations was the result in [2] that dc(perm3) = 7, in particular Grenet’s representation [11] for perm3:
perm3(y) = det7 0 0 0 0 y3 3 y23 y13 y11 1 y12 1 y13 1 y2 2 y21 0 1 y2 3 0 y12 1 0 y23 y22 1 , (9)
is optimal. We sought to understand (9) from a geometric perspective. A first observation is that it, and more gen-erally Grenet’s representation for permmas a determinant of size 2m− 1 is equivariant with respect to about half the symmetries of the permanent. In particular, the optimal ex-pression for perm3 is equivariant with respect to about half its symmetries.
To explain this observation, introduce the following nota-tion. Write Matm(C) = Hom(F, E) = F∗⊗E, where E, F = Cm. This distinction of the two copies of Cmclarifies the ac-tion of the group SL(E) × SL(F ) on Hom(F, E). This acac-tion is (A, B).x = AxB−1, for any x ∈ Hom(F, E) and (A, B) ∈ SL(E) × SL(F ). Let TSL(E)⊂ SL(E) consist of the diagonal matrices and let N (TSL(E)) = TSL(E)o Sm⊂ GL(E) be its normalizer, where Sm denotes the group of permutations on m elements. Similarly for TSL(F ) and N (TSL(F )). Then Gpermm ' [(N (TSL(E)) × N (TSL(F )
))/C∗] o Z2, where the embedding of (N (TSL(E))×N (TSL(F )
))/C∗in GL(Hom(F, E)) is given by the action above and the term Z2corresponds to transposition.
The following refinement of Theorem 1 asserts that to get an exponential lower bound it is sufficient to be equivariant with respect about half the symmetries of the permanent.
Moreover, Grenet’s determinantal representation of permm is equivariant with respect to N (TSL(E)) and has size 2m−1.
We now explain Grenet’s expressions from a representation-theoretic perspective. Let [m] := {1, . . . , m} and let k ∈ [m]. Note that SkE is an irreducible SL(E)-module but it is is not irreducible as an N (TSL(E))-module. For exam-ple, let e1, . . . , em be a basis of E, and let (SkE)reg de-note the span ofQ
i∈Iei, for I ⊂ [m] of cardinality k (the space spanned by the square-free monomials, also known as the space of regular weights): (SkE)reg is an irreducible N (TSL(E))-submodule of SkE. Moreover, there exists a unique N (TSL(E))-equivariant projection πk from SkE to (SkE)reg.
For v ∈ E, define sk(v) : (SkE)reg→ (Sk+1E)regto be mul-tiplication by v followed by πk+1. Alternatively, (Sk+1E)reg is an N (TSL(E))-submodule of E⊗(SkE)reg, and sk: E → (SkE)∗reg⊗(Sk+1E)reg is the unique N (TSL(E))-equivariant inclusion. Let IdW : W → W denote the identity map on the vector space W . Fix a basis f1, . . . , fmof F∗.
Proposition 1. The following is Grenet’s determinantal representation of permm. Let Cn=Lm−1k=0(SkE)reg, so n = 2m− 1, and identify S0E ' (SmE)reg. Set
Λ0= m−1 X k=1 Id(SkE)reg and define ˜ A = Λ0+ m−1 X k=0 sk⊗fk+1. (10) Then (−1)m+1perm
m= detn◦ ˜A. To obtain the permanent exactly, replace Id(S1E)
reg by (−1) m+1Id
(S1E)
reg in the for-mula for Λ0.
In bases respecting the block decomposition induced from the direct sum, the linear part, other than the last term which lies in the upper right block, lies just below the diagonal blocks, and all blocks other than the upper right block and the diag-onal and sub-diagdiag-onal blocks, are zero.
Moreover N (TSL(E)) ⊆ ¯ρA(GA).
2.3
An equivariant representation of the
per-manent
We now give a minimal equivariant determinantal repre-sentation of permm. By Theorem 1, its size is 2mm − 1. For e⊗f ∈ E⊗F∗, let Sk(e⊗f ) : (SkE)reg⊗(SkF∗
)reg → (Sk+1E)reg⊗(Sk+1
F∗)regbe multiplication by e on the first factor and f on the second followed by projection into
(Sk+1E)reg⊗(Sk+1F∗)reg. Equivalently, Sk: (E⊗F∗) → ((SkE)reg⊗(Sk F∗)reg)∗⊗(Sk+1E)reg⊗(Sk+1 F∗)reg
is the unique N (TSL(E)) × N (TSL(F )) equivariant inclusion.
Proposition 2. The following is an equivariant determi-nantal representation of permm: Let
Cn= ⊕m−1k=0(S k
E)reg⊗(SkF∗)reg,
so n = 2mm−1 ∼ 4m. Fix a linear isomorphism S0E⊗S0F∗ ' (SmE)reg⊗(Sm F∗)reg. Set Λ0= m−1 X k=1 Id(SkE)reg⊗(SkF∗)reg and define ˜ A = (m!)n−m−1 Λ0+ m−1 X k=0 Sk. (11) Then (−1)m+1perm
m = detn◦ ˜A. In bases respecting the block structure induced by the direct sum, except for Sm−1, which lies in the upper right hand block, the linear part lies just below the diagonal block.
2.4
Determinantal representations of quadrics
It will be instructive to examine other polynomials deter-mined by their symmetry groups. Perhaps the simplest such is a nondegenerate quadratic form.
Let Q =Psj=1xjyj∈ S2
C2s∗be a non-degenerate quadratic form in 2s variables (such is equivalent toP2su=1zu2 under a change of basis). The polynomial Q is characterized by its symmetries. By [16], if s ≥ 3, the smallest determinantal representation of Q is of size s + 1: ˜ A = 0 −x1 · · · −xs y1 1 . . . . .. ys 1 . (12)
This representation is equivariant with respect to O(s) ⊂ GQ = O(2s) (where the inclusion of O(s) in GL(s+1)2 is first into GLs+1 acting on the last s basis vectors and then into GL(s+1)2 acting by conjugation) and there is no size s + 1 determinantal representation equivariant with respect to GQ. However, Example 1 shows there is a size 2s + 1 determinantal representation equivariant with respect to GQ.
Proposition 3. Let Q ∈ S2CM ∗be a nondegenerate quadratic form, that is, a homogeneous polynomial of degree 2. Then
edc(Q) = M + 1.
2.5
Determinantal representations of the
de-terminant
Although it may appear strange at first, one can ask for determinantal representations of detm. In this case, to get an interesting lower bound, we add a regularity condition:
Definition 4. Let P ∈ Sm
Call the minimal size of a regular determinantal represen-tation of P the regular determinantal complexity of P and denote it by rdc(P ). Let erdc(P ) denote the minimal size of a regular equivariant determinantal representation of P .
von zur Gathen [23] showed that any determinantal repre-sentation of permm or a smooth quadric is regular. In con-trast, the trivial determinantal representation of detmis not regular; but this representation is equivariant so edc(detm) = m.
Theorem 3. erdc(detm) = 2mm − 1 ∼ 4m .
As in the case of the permanent, we can get an exponential lower bound using only about half the symmetries of the determinant.
Theorem 4. Let ˜Am : Matm(C) −→ Matn(C) be a reg-ular determinantal representation of detmthat is equivariant with respect to SL(E). Then n ≥ 2m− 1.
Moreover, there exists a regular determinantal representa-tion of detmthat is equivariant with respect to SL(E) of size 2m− 1.
Remark 1. Normally when one obtains the same lower bound for the determinant as the permanent in some model it is discouraging for the model. However here there is an important difference due to the imposition of regularity for the determinant. We discuss this in further below Question 3.
We now introduce notation to describe the regular determi-nantal representation of detm that is equivariant with re-spect to SL(E) of size 2m− 1 mentioned in Theorem 4. Observe that (SkE)reg ⊂ E⊗k
is isomorphic to the skew-symmetric tensors ΛkE ⊂ E⊗k
as a TSL(E)-module but not as an Sm-module.
Write Matm(C) = E⊗F∗. Let f1, . . . , fm be a basis of F∗. Let exkdenote exterior multiplication in E:
exk : E −→ (ΛkE)∗⊗(Λk+1 E) v 7→ {ω 7→ v ∧ ω}.
Proposition 4. The following is a regular determinantal representation of detm that is equivariant with respect to SL(E). Let Cn= m−1 M j=0 ΛjE, so n = 2m− 1 and
End(Cn) = ⊕0≤i,j≤m−1Hom(ΛjE, ΛiE).
Fix an identification ΛmE ' Λ0E. Set Λ0= m−1 X k=1 IdΛkE, and ˜ A = Λ0+ m−1 X k=0 exk⊗fk+1. (13)
Then detm= detn◦ ˜A if m ≡ 1, 2 mod 4 and detm= − detn◦ ˜A if m ≡ 0, 3 mod 4. In bases respecting the direct sum, the linear part, other than the last term which lies in the upper right block, lies just below the diagonal blocks, and all blocks other than the upper right, the diagonal and sub-diagonal are zero.
Note the similarity with the expression (10). This will be useful for proving the results about the determinantal rep-resentations of the permanent.
When m = 2 this is 0 −y2 2 y21 y11 1 0 y1 2 0 1
agreeing with our earlier calculation of a rank four quadric. Note the minus sign in front of y2
2 because ex(e2)(e1) = −e1∧ e2.
For example, ordering the bases of Λ2
C3 by e1∧ e2, e1 ∧ e3, e2∧ e3, the matrix for det3 is
0 0 0 0 y33 −y32 y31 y11 1 y21 1 y1 3 1 −y2 2 y21 0 1 −y2 3 0 y21 1 0 −y2 3 y22 1 .
We now give a regular equivariant determinantal represen-tation of detm. Let EXk denote the exterior multiplication
EXk : E⊗F∗−→ (Λk
E⊗ΛkF∗)∗⊗(Λk+1
E⊗Λk+1F∗) e⊗f 7→ {ω⊗η 7→ e ∧ ω⊗f ∧ η},
Proposition 5. The following is an equivariant regular determinantal representation of detm. Let
Cn= m−1 M j=0 ΛjE⊗ΛjF∗, so n = 2mm − 1 ∼ 4m and
End(Cn) = ⊕0≤i,j≤mHom(ΛjE⊗ΛjF∗, ΛiE⊗ΛiF∗). Fix an identification ΛmE⊗ΛmF∗' Λ0
and define ˜ A = (m!)n−m−1 Λ0+ m−1 X k=0 EXk. (14)
Then (−1)m+1detm= detn◦ ˜A.
Comparing Theorems 1 and 3, Theorems 2 and 4, Propo-sitions 1 and 4 and PropoPropo-sitions 2 and 5, one can see that detmand permmhave the same behavior relatively to equiv-ariant regular determinantal representations. This prompts the question: What is the regular determinantal complexity of the determinant? In particular:
Question 3. Let rdc(detm) be the smallest value of n such that there exist affine linear maps ˜Am : Cm2 → Cn2 such that
detm= detn◦ ˜Am and rank ˜A(0) = n − 1. (15) What is the growth of rdc(detm)?
In [12] it is shown that rdc(detm) grows at most like O(m4). If one could prove a lower bound on rdc(detm) of m2log(m) or better, and transfer it to the permanent via the Howe-Young duality functor [1], which guided the proofs of the permanental cases in this article, this would be a significant improvement of the state of the art for dc(permm). (This functor also enabled the computation of the linear strand of the minimal free resolution of the ideal generated by sub-permanents in [7].) Because of the symmetries of detn, such a bound might be easier than determining the growth of dc(permm).
3.
OVERVIEW OF THE PROOFS
Let Λn−1∈ Matn(C) be the matrix with 1 in the n − 1 last diagonal entries and 0 elsewhere. Any determinantal repre-sentation ˜A of P of size n with rank( ˜A(0)) = n − 1 can be transformed (by multiplying on the left and right by con-stant invertible matrices) to a determinantal representation of P satisfying ˜A(0) = Λn−1.
The following group plays a central role in the study of reg-ular equivariant determinantal representations:
Gdetn,Λn−1= {g ∈ Gdetn| g · Λn−1= Λn−1}. Let H ⊂ Cndenote the image of Λn−1and `1∈ Cn
its ker-nel. Write `2 for `1 in the target Cn. Then Matn(C) = (`1⊕ H)∗⊗(`2⊕ H). Let transp ∈ GL(Matn(C)) denote the transpose.
Lemma 1. The group Gdetn,Λn−1 is {M 7→λ 0 v g Mλ −1 wT 0 g−1 | g ∈ GL(H), v ∈ H, wT ∈ H∗, λ−1= det(g)} · htranspi.
Let P ∈ SmCM ∗ be either a quadric, a permanent or a determinant. Say a regular representation ˜A is equivariant
with respect to some G ⊆ GP. We may assume that ˜A(0) = Λn−1.
The first step consists in lifting G to GA. More precisely, in each case we construct a reductive subgroup ˜G of GA such that ¯ρA : ˜G −→ G is finite and surjective. In a first reading, it is relatively harmless to assume that ˜G ' G. Then, using Malcev’s theorem (see, e.g. [14]), after possi-bly conjugating ˜A, we may assume that ˜G is contained in (GL(`2) × GL(H)) o Z2. Up to considering an index two subgroup of ˜G if necessary, we assume that ˜G is contained in GL(`2) × GL(H).
The starting point is Schur’s lemma, which restricts non-zero G-module maps between irreducible G-modules to iso-morphic modules. In the case G = GP (respectively G ⊂ GP), then CM is an irreducible G-module (resp. we decom-pose it into a direct sum of irreducible G-modules). Write Matn(C) = (`1⊕ H)∗⊗(`2⊕ H), where H ⊂ Cn is a hyper-plane and the `j are lines.
In the two permanent cases CM is respectively E⊕m as a GL(E)-module and E⊗F as a GL(E) × GL(F )-module Write Matn(C) =` ∗ 1⊗`2 H∗⊗`2 `∗1⊗H H∗⊗H , Λn−1=0 0 0 IdH . If m ≥ 2 the `∗1⊗`2 coefficient of ˜A has to be zero. Then, since P 6= 0, the projection of A(CM) on the first column `∗1⊗H ' H has to be non-zero. We thus have a G-submodule H1 ⊂ H isomorphic to an irreducible submodule of CM (in the permanent cases, respectively E = Cmor E⊗F = Cm2). A similar argument shows that there must be another irre-ducible G-submodule H2⊂ H such that an irreducible sub-module of CM appears in H∗1⊗H2.
For example, in the SL(E)-equivariant determinant case, H1 must be isomorphic to E, so H2 must be such that E ⊂ E∗⊗H1. The only two SL(E)-modules which work are H1= S2
E or H1 = Λ2E. In either case, as long as m > 3, we must have E ⊂ H∗1⊗H3 which implies H3 is an irreducible submodule of E⊗3, the smallest of which is Λ3E. Continuing along the minimal path, one gets the sum of exterior powers as in Proposition 4.
In each case, we construct a sequence of irreducible sub- ˜ G-modules Hkof H satisfying very restrictive conditions. This allows us to get our lower bounds.
To prove the representations ˜A actually compute the poly-nomials we want, in the case G = GP, we first check that GP is contained in the image of ¯ρA. Since P is characterized by its symmetries, we deduce that detn◦ ˜A is a scalar mul-tiple of P . We then specialize to evaluating on the diagonal matrices in Matm(C) to determine this constant, proving in particular that it is non-zero.
4.
ACKNOWLEDGMENTS
Geometry at the Simons Institute for the Theory of Com-puting, UC Berkeley. Most of the work was done when the Landsberg was a guest of Ressayre and Pascal Koiran. Landsberg thanks his hosts as well as U. Lyon and ENS Lyon for their hospitality and support. We thank: Christian Iken-meyer for useful discussions about equivariant determinan-tal presentations of the permanent and Grenet’s algorithm, Josh Grochow, Christian Ikenmeyer and Shrawan Kumar for useful suggestions for improving the exposition, J´erˆome Germoni for mentioning the existence of the spin symmetric group, Michael Forbes for suggesting Example 3, and Avi Wigderson for suggesting Example 4. Landsberg supported by NSF grant DMS-1405348. Ressayre supported by ANR Project (ANR-13-BS02-0001-01) and by Institut Universi-taire de France.
5.
REFERENCES
[1] K. Akin and J. Weyman. Primary ideals associated to the linear strands of Lascoux’s resolution and syzygies of the corresponding irreducible representations of the Lie superalgebra gl(m|n). J. Algebra, 310(2):461–490, 2007.
[2] J. Alper, T. Bogart, and M. Velasco. A lower bound for the determinantal complexity of a hypersurface. ArXiv e-prints, May 2015.
[3] D. A. M. Barrington, R. Beigel, and S. Rudich. Representing Boolean functions as polynomials modulo composite numbers. Comput. Complexity, 4(4):367–382, 1994. Special issue on circuit complexity (Barbados, 1992).
[4] J.-Y. Cai. A note on the determinant and permanent problem. Inform. and Comput., 84(1):119–127, 1990. [5] J. y. Cai, X. Chen, and D. Li. Quadratic lower bound
for permanent vs. determinant in any characteristic. Computational Complexity, 19(1):37–56, 2010.
[6] P. Comon. Tensor decompositions, state of the art and applications. In J. G. McWhirter and I. K. Proudler, editors, Mathematics in Signal Processing V, pages 1–24. Clarendon Press, Oxford, UK, 2002.
arXiv:0905.0454v1.
[7] K. Efremenko, J. M. Landsberg, H. Schenck, and J. Weyman. On minimal free resolutions and the method of shifted partial derivatives in complexity theory. ArXiv e-prints, Apr. 2015.
[8] I. Fischer. Sums of Like Powers of Multivariate Linear Forms. Math. Mag., 67(1):59–61, 1994.
[9] G. Frobenius. ¨Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen. Sitzungsber Deutsch. Akad. Wiss. Berlin, pages 994–1015, 1897. [10] D. G. Glynn. The permanent of a square matrix.
European J. Combin., 31(7):1887–1891, 2010.
[11] B. Grenet. An Upper Bound for the Permanent versus Determinant Problem. Theory of Computing, 2014. Accepted.
[12] C. Ikenmeyer and J. Landsberg. Ranks of determinantal representations. preprint. [13] J. Landsberg. Geometry and complexity theory.
preprint available at
http://www.math.tamu.edu/∼jml/simonsclass.pdf. [14] J. Landsberg and N. Ressayre. Permanent v.
determinant: an exponential lower bound assuming symmetry and a potential path towards valiant’s
conjecture. arXiv:1508.05788, 2015.
[15] M. Marcus and H. Minc. On the relation between the determinant and the permanent. Illinois J. Math., 5:376–381, 1961.
[16] T. Mignon and N. Ressayre. A quadratic bound for the determinant and permanent problem. Int. Math. Res. Not., (79):4241–4253, 2004.
[17] K. D. Mulmuley. On P vs NP, geometric complexity theory, and the flip I: a highˆa ˘A¸Slevel view. Technical Report TRˆa ˘A¸S2007ˆa ˘A¸S09, Computer Science Department, The University of Chicago, july 2007. Also available as arXiv:0709.0748.
[18] K. D. Mulmuley and M. Sohoni. Geometric complexity theory. I. An approach to the P vs. NP and related problems. SIAM J. Comput.,
31(2):496–526 (electronic), 2001.
[19] K. D. Mulmuley and M. Sohoni. Geometric
complexity theory. II. Towards explicit obstructions for embeddings among class varieties. SIAM J. Comput., 38(3):1175–1206, 2008.
[20] K. Ranestad and F.-O. Schreyer. On the rank of a symmetric form. J. Algebra, 346:340–342, 2011. [21] L. G. Valiant. Completeness classes in algebra. In
Proc. 11th ACM STOC, pages 249–261, 1979. [22] L. G. Valiant. Short monotone formulae for the
majority function. J. Algorithms, 5(3):363–366, 1984. [23] J. von zur Gathen. Permanent and determinant.
Linear Algebra Appl., 96:87–100, 1987.
