An algorithm for recognising the exterior square of a multiset Catherine Greenhill School of Computer Studies University of Leeds Leeds LS2 9JT United Kingdom 5th January 2000 Abstract The exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of el- ements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algo- rithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further com- ments are made about the motivating case of multisets of eigenvalues of matrices. 1 Introduction The exterior square of a vector space is a well known and important construction with applications in various areas of mathematics. In this paper the exterior square of a multiset is defined. This definition arises in a natural way, as it describes the relationship between the eigenvalues of a matrix X and the eigenvalues of the matrix representing the action of X on the exterior square of the underlying vector space. This relationship is fully determined by the nonzero eigenvalues, which belong to the multiplicative group of the splitting field of the characteristic polynomial of X. Therefore the appropriate generalisation is to multisets of elements of an abelian group. For most of the paper we consider the problem in full generality. Two properties are defined which a multiset may possess; the properties of recog- nisability and involution-recognisability. An algorithm is developed which can deter- mine whether a given multiset is the exterior square of a recognisable or involution- 1 recognisable multiset. This strategy is adapted from that used by Peter Neumann and Cheryl Praeger in their work on the tensor products of multisets [7]. The worst-case complexity of the algorithm is analysed, under the assumption that the elements of the abelian group can be linearly ordered. Let m, n be positive integers such that m 2 and n = m(m 1)=2. Suppose that the input multisets have n elements. The≥ basic operation is taken− to be one multiplication, inversion or comparison of elements of the abelian group. Denote by C(2) the cost of finding square roots in the abelian group. Then the worst-case complexity of the algorithm is 2 3 O n m C(2) + n m log(n) : Heuristic arguments are outlined which suggests that the improved bound 2 O n C(2) + n log(n) is more appropriate for the majority of inputs. The algorithm cannot recognise the exterior square of a multiset which is neither recognisable nor involution-recognisable. In Section 6 we show that the proportion of multisets which are neither recognisable nor involution-recognisable is small when the abelian group is finite but large. The theoretical results and heuristic arguments are illustrated by the results of practical tests, which are presented in Section 7. Finally, in Section 8 we consider the special case which motivated the general defi- nition of the exterior square of a multiset; namely, multisets of eigenvalues of matrices. We show how the multiset algorithm can be used to recognise the exterior square of certain matrices over finite fields (up to conjugation). This might help us to solve the open problem of recognising the special linear group in its exterior square action. 2 Notation and preliminary results First we review the definition of a multiset. Let Θ be any set. The set of all multisets r of size r with elements in Θ is denoted by Θf g and is defined by r r Θf g = Θ =Sym(r); r r where the symmetric group Sym(r) acts on Θ by permuting entries. If (!1;:::;!r) Θ r r 2 then denote its image in Θf g by !1;:::;!r . By convention, if ! Θf g then we write f rg 2 ! = !1;:::;!r . Suppose that ! Θf g and g Θ. Let mult(g; !) denote the multiplicityf of g ing !, defined by 2 2 mult(g; !) = i : g = !i : j f g j Write g ! if mult(g; !) 1. Say that ! is multiplicity-free if mult(g; !) 1 for all g Θ. The2 multiset union,≥ multiset intersection and multiset difference operations≤ can all2 be defined in terms of multiplicities, as follows. Let mult(g; ! !0) = mult(g; !) + mult(g; !0); [ mult(g; ! !0) = min mult(g; !); mult(g; !0) ; \ f g mult(g; ! !0) = max 0; mult(g; !) mult(g; !0) n f − g 2 r r for all g Θ and all !; !0 Θf g. Let k be a positive integer and ! Θf g. Define the 2 kr 2 2 multiset k! Θf g as follows: 2 mult(g; k!) = k mult(g; !): r We might wish to define a multiset using some property defined on Θ. Given ! Θf g, write P 2 !0 = g ! : (g) f 2 P g to denote the multiset defined by 0 if g ! or not (g); mult(g; ! ) = ( 0 mult(g; !) otherwise62 : P Finally, if Θ is a finite set then r Θ + r 1 Θf g = j j − ! (1) j j r for r 1. ≥ 2.1 The exterior square of a multiset Suppose that A is a set which supports a commutative multiplication operation. Let m 2 a be an element of Af g. The exterior square of a, denoted by a^ , is the multiset aiaj : 1 i < j m : f ≤ ≤ g n For the remainder of this paper let n = m(m 1)=2. Let b be an element of Af g. − 2 Then a is said to be an exterior square root of b if b = a^ . To justify the use of the term \exterior square" in this context, we show how this definition relates to the most well-known exterior square construction, the exterior square of a vector space. Let X be an m m matrix over a field F and let α be the multiset of eigenvalues of X. It is not × 2 difficult to prove that α^ is the multiset of eigenvalues of the matrix which represents the m action of X on the exterior square of the underlying vector space. Note that α Kf g 2 n 2 and α^ Kf g, where K is the splitting field of the characteristic polynomial of X over 2 2 F . Clearly α^ is determined by the multiset of nonzero elements in α. For this reason, we consider multisets of abelian group elements. For the remainder of the paper let A be an abelian group. n Suppose that we were given a multiset b Af g and told that b has an exterior square root. Suppose that we were also given a2 map : i; j : 1 i < j m 1; : : : n ff g ≤ ≤ g ! f g such that aiaj = b ( i;j ) for 1 i < j m. Then, writing the abelian group A f g ≤ ≤ additively, we have n linear equations in m unknowns a1; : : : ; am. We can solve the 3 system to find the entries of a using elementary linear algebra over the integers. Here multiplying an equation by a positive integer r corresponds to raising each side of the equation to the rth power and multiplying an equation by 1 corresponds to inverting each side of the equation. Dividing an equation by a positive− integer r corresponds to taking rth roots of both sides of the equation in A, and so is only allowed when both n sides have an rth root in A. Of course, when given an element of Af g we do not in general have access to the helpful map , and simply testing all possible maps does not lead to an efficient algorithm as there are n! of them. (Note that this approach does provide an algorithm, albeit a highly inefficient one, which applies when the algorithm described in this paper does not: namely, for finding exterior square roots which are neither recognisable nor involution recognisable.) 1 3 It is easy to construct exterior square roots of elements of Af g and Af g. Let 1A denote the identity element of A. Then an exterior square root of b1 is given by 3 f g 1A; b1 for all b1 A. Let b be an element of Af g. If b has an exterior square root a then,f withoutg loss2 of generality, b1 = a1a2; b2 = a1a3; b3 = a2a3: 1 2 3 1 Therefore b1b2b3− = a1 . Given b Af g, if b1b2b3− has no square root then b has no 2 1 exterior square root. Otherwise, let w be a square root of b1b2b3− . Then the multiset 1 1 w; w− b1; w− b2 is an exterior square root of b. For the remainder of the paper assume thatf m 4. g ≥ 2.2 Preliminaries To close this section, we make a few more definitions and establish some preliminary r r results. If a Af g and τ A, define the multiset τa Af g as follows: 2 2 2 1 mult(g; τa) = mult(τ − g; a) for all g A. Observe that if τ is any element of A such that τ 2 = 1 then 2 2 2 (τa)^ = a^ (2) m m 1 for all a Af g. For a Af g let aa− denote the multiset 2 2 1 ai aj− : 1 i = j m (3) n ≤ 6 ≤ o 2 2 with m(m 1) elements, and let a^ a−∧ denote the multiset − 1 aiaj(akal)− : 1 i < j m; 1 k < l m; i; j; k; l distinct (4) n ≤ ≤ ≤ ≤ o m 2 2 with 6 4 elements.