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Linear and Multilinear Algebra

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Permanents and determinants with genericnoncommuting entries

Joy P. Fillmore & S. Gill Williamson

To cite this article: Joy P. Fillmore & S. Gill Williamson (1986) Permanents and determinantswith generic noncommuting entries, Linear and Multilinear Algebra, 19:4, 321-334, DOI:10.1080/03081088608817727

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Lirlrar m7d kf~rltilmear 4lgebrrc 1986. Vol 19. pp 121-334 0308-1087 86 19040321 $20 00 0 c 1986 Gordon and Breach Sc~ence Publishers S A Prrnted rn the Unlted States of Amer~ca

Permanents and Determinants with Generic Noncommuting Entries JOY P. FILLMORE and S. GILL WILLIAMSON University of California, San Diego, La Jolla, California 92093

(Received Alryust 12. 19x5)

This paper studies some basic comb~natorial properties of matrix functions of generic matrices. A generic matrix is one with entries from a free associative algebra, over a field, and on a finite set of non-commuting variables (i.e. a tensor algebra). The principal tools are shuffle products. Generic column and row permanents are defined and analogs of the Laplace and Cauchy-Binet theorems are derived in terms of shuffles. In this setting, the generic permanents include as special cases all of the classical matrix functions: Schur matrix functions, determinants, and permanents. 1980 Mathematics Classification 05.15. Keywords: Shuffle product, generic matrix functions, minor expansions. Laplace Expansion Theorem, Cauchy-Binet Theorem, permanents, determinants, tensor algebra, matrices with non-commuting entries.

1. INTRODUCTION

We record here some results in combinatorial matrix theory. These results are motivated by studies [ I , 81, where a need for computational techniques in certain areas of linear and multilinear algebra arose that were not a consequence of the classical framework. In particular, we deal here with determinants of matrices with noncommuting entries such as arise classically from linear algebra over general rings. The usual determinants of such matrices necessarily take on values which commute and are multjlinear functions of both rows and columns. In many applications, however, these determinants do more than just provide a number. They serve to organize and provide essentially combinatorial insights into calculations. It is this latter aspect that admits a generalization to the noncommuting case by not demanding all the usual properties of determinants. In this paper, a generalization of the determinant is described. It is remarkable that the appropriate generalizations of the classical identities of Laplace and Cauchy-Binet

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322 I 1' F I L L M O R E AND s G WILLIAMSON

remain valid and have even simpler proofs. These results, which we describe in detail in this paper, lend insight into the classical case by bringing combinatorial aspects to the forefront and are, in this sense, closely rclated to the approach of Marcus [6] .

The key to defining something like a determinant with noncommuting entries is the need to be able to multiply noncommuting monomials and have the variables arranged in a specified order. This is done by means of a shuffle product. When this product is used on generic variables, the positions of thc variables themselves record how they were multiplied. With this bookkeeping, permanents of generic matrices are defined and their Laplace expansion theorems obtained in Section 2. Since generic variables keep track of their multiplication, the usual signs that are introduced in passing from permanents to determinants can be replaced by arbitrary representations, leading to Laplace expansion theorems for generalizations of the Schur matrix functions [7, 81. This is carried out in Section 3. In Section 4, analogues of the Cauchy-Binet theorem are obtained in this setting.

Shuffle products and related ideas occur in a similar view in ring theory, algebraic topology, in the theory of algorithms. to mention a few, and combinatorics. The latter touches most closely the viewpoint of this paper [I, 3, 4, 51.

2. GENERIC LAPLACE EXPANSIONS

Let W be a finite set. We consider Was noncommuting variables I n the free associative k-algebra k ( W ) over the field k. If kW denotes the free vector space wtth W as basis, then k(W) may be identified with the full tensor algebra @ kN : the monomial a,a, . . . a, of degree p is identified with the homogeneous tensor u , O a, O.. .@I a,.

Let m be the ordered set { 1 ,2 , . . . , m ) of integers. For an (ordered) subset I c T I , let u, be the monomial n,,, a,, where the order of the variables is that o f l . For I c m, 7 = m - I is the complement of I with its order as subset.

2.1. Definitiorl (Shuffle product) Let a = ( I , . . . a, and b = h , . . . b, be two monon~ials in W. Let I c in + n have cardinality jlj = m, so f =

m + t7 - I has cardinality n. Define the I-shuffle a !lJ h of a and h to be --

c = c , c , . . . cm+,, where c, = a and ci = h. The shuflllr prodzrct a uu b of a

and b is the sum of all I -shuffles u 111 h over all I E nl + n with 11) = rrz. - --

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PERMANENTS AND DETERMINANTS 323

2.2. Exanzple Take cr = a,a,. b = h , h,b3, and I = (2,4). Then c = clc2c,c,cs gives that c, = c,c, is a = a,a, and c ,+, = c,c,c, is b,b,h3, so u A h = b,u,b,a,b,. And a uu b = a ,~~ ,b ,h ,b , + a l h l ~ 2 b 2 h 3 + a,b,h,a3h3 + a,b,h,b,u, + b,a,a,b,b, + blulb,a,h3 + b,a,b,b,n, + h,h,a,u2h3 + b,b,u,b,u, + b,b,b,u,a,.

2.3. Revlark The vector space underlying the algebra k(W) together with the shume product w is an associative algebra called the shuffle algebra of W. It is generated by all finite shume products L I , uh a, uu . . . UL a,, a , E W , and is isomorphic to the full symmetric algebra ~ ( k " ) as well as the algebra k[W] of commuting polynomials in the variables in the set W.

The classical determinant expansion theorems of Laplace and Cauchy-Binet do not depend on the entries of the determinant being commutative procided one has the correct notion for the determinant for noncommuting variables ard a means of keeping track of the variables. This will be achieved by introducing generic. matrix functions and specializing carefully at the appropriate times.

2.4. Drfinitiorl (Generic permanents) Let A = ( ( I , , ) be an m x m matrix with entries in W. Define

to be the generic row and column permanents of A, respectively. S , is the set of permutations of m.

2.5. Noratiort Let A = (a,) be an nz x n matrix. If g E pm, the set of all functions from m to t, then A, , is the m x nz matr~x having columns g(l), . . . , g(m) from A. Writing A( i , j ) = a,,, one has A,,(7, J ) = a ,,,.

An rn x n matrix A = (a,) with entrles nzn distinct elements of W will be called a generic matrix over W . In this notation, we have:

2.6. LEMMA Let u , = I;=, a i j , i = 1 ,2 , . . . , m be generic matrix A = ( a i j ) Then the shuffle product

the row s u m of a of the row sums is

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324 J. P. FILLMORE A N D s G WILLIAMSON

expressible as

(1) u , UU...IJU urn = 1 PER, ( A , , ) . QE"

ProqJ For distinct generic variables sij one has

since the left-hand side is the sum of monomials x , j , . . . xmjm and the j's define the g. Replace . x , ~ by aOi,, and sum on cr E S, to obtain

and this is the sum of monomials a ,,,,, . . . a which is evidently 11, UU . ' ' U u U,,, .

2.7. Norur ion The following notation will be useful in describing the generic Laplace expansion and Cauchy-Binet theorem.

For an In x n matrix A, and I E in and J E n nonempty, let A [ I I J ] be the submatrix of A having rows indexed by I and columns indexed by J . Let A ( 1 I J ) = ~ [ i l j ] , i = nz - I and j = i t - J, be the matrix obtained by deleting rows indexedby 1 and colu& indexed by J. The mixed notation A [ I I J ) and A ( I I J ] have the obvious meanings.

For I and J subsets of m. let Sm(I, J ) be the set of all permutations of rt1

which map I to J :

Note

(a) A permutation o of S,(I, J ) is completely determined by its restrictions rrll and rrll to 1 and its complemeut f. Thus if Ill = p , lSm(l.J)l = p! (in - p ) ! .

(b) If I , is a fixed nonempty subset of m. then Sm is the disjoint union

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PERMANENTS AND DETERMINANTS

Likewise for fixed nonempty Jo E m,

These disjoint unions are reflected in m ! = p! (m - p ) ! . (3 2.8. THEOREM (Laplace expansion for generic permanents) Let A = (a,,) be a generic m x m matrix (entries are distinct elernents o f W) .

( 1 ) For I, E nz fixed and (JI = (Io(.

(2) For Jo G m fixed and (11 = lJol,

(3) For Jo c m ,fixed and Ill = IJ,I,

(4) For 1, m fixed and (J( = (l,l,

Proof Consider the first formula (1). By definition of the 1,-shuffle, one has for any o E Sm:

m

" i ,o i = fl u i , n i u 11 cli ,u i . i = I i t l , itl,

Now, if o E Sm(IO, J ) , then I l iEIo is a summand for PER ,(AIIo(J]) and likewise the second factor for PERs(A(I0IJ)). By note (a) above, we have

By note (b) above, summing over J c_ m gives PER , ( A ) provlng formula (1). Forn~ulas (2), (3), and (4) are proved in a similar fashion.

Classically, formula (1) would be called the expansion by rows indexed by 10.

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326 J P FILLMORE AND S. G WILLIAMSON

Let x,. . . . , xrn be noncommuting variables each of which commutes

with every element W. Thus X = (a . ' 1 is a generic diagonal

.urn matrix. In a generic t n x n matrix A = (u i j ) , we may replace c l i j by . u , r~ ,~ = uij.u, in any formula. For example,

PER ,(.AX) = 1 n ui,ni xrii " E S , i= 1 i= l

The effect is twofold: n xGi keeps a record of the permutations a in the summands, and it also plays a role like that of the signs distinguishing a determinant from a permanent. Observe also PER,(AX) is a polynomial in the set of noncommuting variables {x,, . . . , x,}. The permanent PERs(AX) will be called the generic row nzurrix ,fimction. Likewise,

rn rn

P E R , ( X A ) = 2 n umj , j n .Ycj m ~ S , , , j - 1 j L 1

is the generic colirrnti mrrtrir ,firnction. It is well known that the classical Laplace expansion theorem is a

consequence of the associativity of the exterior product in a Grassmanian algebra [h] . In a similar fashion, the Laplace expansion theorem for generic permanets and its consequences for generic matrix functions, is a consequence of the associativity of the shuffle product in the shume algebra. We now explain the connection in this latter setting.

The key is several applications of Lemma 2.6, but we require a technical observation. Let I, be a fixed subset of m. For each J G m with l J J = JI,I, let b, denote the order preserving bijection from I, to J . If h E p'". then h 2 h, E 1 1 ' ~ . Thus / I hJ sends i, t o h ( j , ), where i , < i, < . . . < i, and j , < j2 < . . . < j , are the ordered elements of I , and J , respectively. Likewise, h j is the order-preserving bijection from 7, = m - I, to j and h : b j € ~ ' ( 1 .

The observation : Given I, a i d J , lJI = I1,I, each h E urn gives rise to the pair of functions h : b, E nIu a n d h b j E and, for fixed J , every pair of functions from !'ll and !Tij is so obtained:

( ( . f i g ) : ,~'E!I'(', g ~ n i ' ) = { ( h o h J , h 2 h j ) : h € n r n ) .

Indeed, this is a "change of indices" from I, to J. Recalling Notation 2.7, we have

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PER MANE^ rs A N D DETERMINANTS 327

2.9. THEOREM Let A be 1111 m x n grnrr ic rnutrix. Then jor each jiwed subset I, c 111,

1 PER, ( A , , ,I = 1 x PERs ( A , , , hJ[lol_nl)lJl PERs (A , , , h,(loln]) hen"' hen" i J c m

Proof Let u , = x:=, a,, be the row sums of A. For any CTE S,,

by (1) of Lemma 2.6. Thus by associativity,

where U_j u i is the shuffle product in the order given by I , and likewise for To. By (2) of Lemma 2.8, applied once to the left side and twice to the right side of the last equation, gives

The desired assertion now follows from the observation preceding the theorem.

Since the matrix A = (u,,) of Theorem 2.9 is generic, the monomials in

are linearly independent over the field k and we may extract the summands for each h in the theorem to obtain a formula for them.

2.10. COROLLARY For uny h E!" and 1, G m, PER, (Alh) = 2 PER, (Al,h h J I I o l ! l ] ) ( J ( PER, (A,, , h j ( l o l n ] ) .

J = m

lJl=lJol

If A is an m x r n syuare matrix and 11 is the identity map, then Corollary 2.10 is just Theorem 2.8(4).

Both 2.9 and 2.10 immediately give expansions for the generic column matrix functions expressing PER, ( X A , , ) and its sums over h E nm in

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3 28 J P. FIL-LMORE A N D S. G WILLIAMSON

terms of sums of

( (XA)l ,h b , [ l O \ ? ] ) l g ( ( X A ) l , h b , ( l O / ! l ] )

over J _c t n and h E 17". Theorem 2.9 and Corollary 2.1 0 have obvious analogs for row permanents PER ,.

3. GENERIC SCHUR MATRIX FUNCTIONS

Classically, one obtains the determinant from the permanent by introducing a plus or minus sign before a summand a , ,,, . . . . - a,,,, , the sign being the value of the alternating character of the permutation a in S,. This choice of signs was extended by Schur [7] to include representations of S , and such an extension may be incorporated into the generic case also.

3.1. Definition (Generic Schur matrix functions) Let M be a representation of the group S, of perturbations of m. For any 1n x n7 matrix A = (a,,) with entries in W, define

to be the generic row and column Schur matrix ,functions of A with respect to M.

When M is a matrix representation of S,, these Schur matrix functions are matrices with entries in k(W).

To obtain a Laplace expansion theorem for Schur matrix functions, we need several preliminary observations.

3.2. Let I L m be of cardinality (I1 = p.

(a) Let c, be the order-preserving bijection from 0 to I. If r 6 S,, define

611 = c, x (., ' . Thus i 6 $,(I, I ) = s,(K 7) and is just

carrying out the permutation x on the p elements of I and fixing pointwise the elenlents of 7.

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PERMANENTS A N D DETERMINANTS 3 29

(b) In a like manner, let (',- be the order-preserving bijection from f(1 = id11

p' - = m - - p - to i. If DES, - p , define ,8 by 4.p.;~'. Then

PE ~ , ( f , i ) = S,(I , I ) is the permutation 0 carried out on the m - p elements of f .

Let I and J be two subsets of m of the same cardinality p. Recall that S,(I, J ) is the set of rr E S , mapping I to J .

i - 1 $11 = c , c,

(a) Define $ by - --,. Then is the unique map of $11 = F j - cT

S , ( I , J ) = ~ , ( f , J ) which preserves order from 1 to J and from f to J.

(b) Note that every element of S,(I. J ) = ~ , ( i , j ) is uniquely Il/i,8 = $,8i with r G S , and f i E S,, - p .

Note The dependency of 2 and on 1 and of Il/ on I and J may be denoted by subscripts: ;,,ti, $ I , J . These subscripts will frequently be omitted.

(c) If M is a representation of S,, then M I ( % ) = M(c , r c ; ' ) gives a representation of S,; r is sent to M(i). If A[l IJ] is the p x p submatrix of A obtained by selecting rows from I and columns from J. then

ntS, ie l

A similar observation holds for r M i ( D ) = M(&fl?f I ) and A[TIJ] =

A (1, J ) .

3.3. THEOREM (Laplace expansion theorem for generic Schur matrix functions) Let A = (u,,) he a yrnerlc m x m m a t r ~ x und M ri

reprewntntion oj S,. For yerwrlc rovj Sthur rncltrru functrow\ orw ha\:

PER: ' (A) = 1 M($,,") PER':J(A[IIJ,] )JIJ P E R : " ( A ( I I J , ) ) .

And s~mllur fornzulas for generic tolutnn Sthur imtrl,: functionf.

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330 J . P. F I L L M O R E AND S. < i WILLIAMSON

Prooj' This proof is an extension of that of 2.8. By the definition of the I,-shuffle and 3.2(b),

where a = ri/iJ. Multiply by M(a) = M ( $ ) M ( ~ ) M ( ~ ) and sum on o E S,(I,, J) to obtain

By note (c) above, the right-hand side is

The proof is concluded by summing over J c m by note 2.7(b). The other formulas are proved in a similar fashion.

3.4. Remark Let A be a generic m x m matrix, X a generic diagonal matrix whose entries commute with those of A, and M a representation of S,. Since x,, . . . .xo, uniquely determines o, we may replace x,, . . . x0, in PER,(AX) by M(a) to obtain P E R ~ ( A ) . Theorem 3.3 is a consequence of Theorem 2.8 for matrices A X and XA once one describes the multiplication rules for the sequences of x's.

In fact, the four formulas for

encompass both sets of formulas.

3.5. Exumplr Let E be the representation of S, which is the alternating character which assigns to o E S, its sign &(a) = rt 1 according as the permutation is even or odd. If I = {i, < . . . < i,) and J = { j , < . . . < j,) in m, then

E ( $ , J ) = ( - 1 ) ~ ' ~ + ~ ' ~ .

In this case, Theorem 3.3(1) applied to matrices A is the Laplace expansion of

det A = PEK',s(A)

by unions and complementary unions with its well-known choice of signs [6]. In the classical case, the entries of A are commuting variables.

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PERMANEhTS 4YD DETERMINANTS 331

4. GENERIC CAUCHY-BINET THEOREMS

The classical expansion theorem of Cauchy Binet [6] has analogs for generic permanents and Schur matrix functions that are of the same style as previous analogies. We begin with a preliminary observation.

The group S , of permutations of m acts on nm by means of the argument off '€ !I"': of' = f ' . K1. The stability subgroup of an element f E - 11" has cardinality

and each orbit of S , contains a unique element of Trim, the set of nondecreasing functions from m to n. Thus a sum of quantities Q ( f ' ) indexed by f G nm may be written as a sum over the disjoint orbits of S,:

4.1. THEOREV (Gener~c Cauchy Blnet) Let A = (a,,) and B = (b,,) be gewri( 111 x 11 m r l n x n~ mutrice\. re\pettu PI\. Then

PER, (AB) = 1 A n PERs (A1,,,-l )Bh,-l,,(j,,j 1. hs+rtm v ( h ) *ES," j = 1

Proof' As in the proof of Lemma 2.6, for distinct generic variables x,, one has

For fixed a G S,, let xir = ai,thr,oi. Then

n ( A B ) ( i , ~ i ) = n A( i , hi)B(hi, oi). I= 1 hen" I = 1

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332 J. 1'. FILLMORE A Y D S. Ci WILLIAMSON

Recall from 2.5, A ,,(i, j j = u ~ , , ~ ~ . Sum on o E S, to obtain

P E R , (AB) = x fl A(i, hi ) PER ,(Bh1). hen" I = 1

By the preliminary observation, we have the first formula. The second formula is proved in a similar fashion.

4.2. Renlilrk As in Section 3, a representation M may be included in the Cauchy Binct theorem. If, in the first formula of the proof of 4.1 we set .xit = ~ i ~ , , h , , , ~ M ( o ) , the succeeding steps lead to PER:. This givcs as corollary of the proof the formula

and the analogous ones for PER,~"(AB).

Supposc now that the entries of an m x m square matrix C' commute. Then we have an equality of products

for any $I E S,. We may apply this observation to the tn x i n matrix R,,, obtained from an T I x m matrix B and h E 11".

for IL E 17"' ' l t l l l fp E S,,,

Proof' As rr ranges over Sm so does (b-'a, so for B = ( h i j ) we have

Since h2 is a representation. the preceding observation gives the formula.

This now yields a Cauchy-Binet theorem where the entries of one matrix commute among themselves but not necessarily with the entries of the other matrix.

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PERMANEKTS AND DETERMINANTS 333

4.4. THEOREM Let A uvld B be generic m x n and n x. nz mutric~s, respecticely with the entries qf'B conzmuting among thernselcrs, and let M he a representation of S,. Then

Proof By 4.2 and the 4.3, PER:i(AB) is m

Replacing 4- ' by d, we recognize the inner sum as PERy(A,,). The analogous formula for PERf(AB) holds when the entries of A

commute, but not necessarily with those of B.

4.5. E.uunzple Let the entries of the m x m square matrix A commute.

(a) If the representation M is thc trivial character assigning the scalar 1 to every o E S,, then

per A = P E R . ~ ( A )

is the classical permanent of A. By commutativity, this row permanent is the same as the column permanent PER,y(A). By 4.3, per A is a symmetric function of the rows or columns.

(b) If the representation M is the alternating character E as in 3.5. then

det A = PERt,(A )

is the classical determinant of A. This row determinant is the same as the column determinant PER; (A) by commutativity. By 4.3, det A is an alternating function of its rows or columns.

Let ?!1'" denote the set of all strictly increasing functions from to c.

4.6. COKOLLAR\~ (Class~cal Cauchy-Binct) Let A and B he m x n and n x nz matrices wi th 011 entries commuting nnzong each other. Then

1 (a) perAB= 1 -perAl,,pcrBh,.

hc',1" \'(hi

(b) de tAB= 1 detAlhdetBh1 h c 0 11

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Proqf' This is an immediate consequence of 4.4 and 4.5 once we observe for (b) that det A , , = 0 by repeated colu~nns unless k E Trl"' is strictly increasing, and in this case ~ ( 1 7 ) = I .

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[I] E. R. Canfield and S. C . Williamson. Hook length products and Cayley operators of classical inbariant theory, Liil. hlultilir~. Alg . 9 (1981). 283-295.

[ 2 ] M. Clausen. Letter place algebra and a characteristic-frec approach to the representation theory of the general lincar and symmerric groups. 1, A ~ I . . in .Molh. 33 (1979) No. 2. Ihl 191.

[3] J., DCsarminian, U. P. S. Kung and G-C. Rota. Invariant theory, Young bitableaux and combinatorics, Adr. ill Mccth. 27 (1978). No. I. 63-92,

[4] P. Doubilet, G-C. Rota and J. Stein, On the foundations of combinatorial thcory I X : Combinatorial methods in invariant theory. Stud. A p p l . Math. 53 (1974) No. 3, 185-216.

[S] A. M. Garsia and J. Remmel. Shuffles of permutations and the Kronccker product, Preprint, Department of Mathematics. UCSD. La Jolla. CA. 1985.

[6] M. Marcus, Fit~icc, D~tiltvr.\ioml l.lr~l/ili,ii~ctr Al~gehrtr. Marcek Dekker, Vol. I , 1973: Vol. 11. 1974.

[7] I. Schur. Ueber cndliche G r ~ ~ p p e n und Hermiteschrn Formen. .Zltr~h. Z. 1 11918). 184-207.

[XI S. G. Willismson. Symmetr) operator.;. polarizationh, and a generalized Capelli identity, I.i~i. Mtrltilit~. Akq. 10 (1981 ). 93-102.

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