- Email: [email protected]
Sets of Uniqueness and Minimal Matrices
208, 444᎐451 Ž1998. JA987496
JOURNAL OF ALGEBRA ARTICLE NO.
Sets of Uniqueness and Minimal Matrices Adolfo Torres-Chazaro ´ Departamento de Matematicas, Uni¨ ersidad Autonoma Metropolitana, Unidad ´ ´ Iztapalapa, Michoacan 09340 Mexico, D.F., Mexico ´ y Purısima, ´ ´
and Ernesto Vallejo* Instituto de Matematicas, Uni¨ ersidad Nacional Autonoma de Mexico, ´ ´ ´ Area de la In¨ . Cient., 04510 Mexico, D.F., Mexico ´ E-mail: [email protected] Communicated by Georgia Benkart Received May 5, 1997
In this note we give an algebraic characterization of sets of uniqueness in terms of matrices with non-negative integer coefficients and prescribed row and column sums, and of the dominance order of partitions or majorization. Our proof uses some identities involving characters of the symmetric group. As an application we describe all sets of uniqueness contained in the box B Ž2, 2, r ., for all r. 䊚 1998 Academic Press
1. INTRODUCTION For a positive integer m let w m x [ 1, . . . , m4 . Given a subset S of the 3-dimensional box B s B Ž p, q, r . [ w p x = w q x = w r x we consider the cardinalities of its slices parallel to the coordinate planes
i s Ž x1 , x 2 , x 3 . g S x1 s i4 ,
1 F i F p,
j s Ž x1 , x 2 , x 3 . g S x 2 s j4 ,
1 F j F q,
k s Ž x1 , x 2 , x 3 . g S x 3 s k4 ,
1 F k F r.
* Supported by DGAPA UNAM IN-103195 and IN-103397. 444 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
Ž 1.
SETS OF UNIQUENESS AND MATRICES
445
The slice ¨ ectors S [ Ž 1 , . . . , p ., S [ Ž 1 , . . . , q ., S [ Ž 1 , . . . , r . are compositions of the cardinality of S. The set S is called a set of uniqueness if it is the only subset of B Ž p, q, r . with slice vectors equal to S , S , S . Sets of uniqueness were introduced in w3x, where a geometric characterization of them was given by the absence of certain configurations in B Ž p, q, r .. Their work can be viewed as an extension of previous work on similar questions in the context of Boolean function theory, switching circuit theory, and game theory Žsee references in w3x.. In this note we give an algebraic characterization of sets of uniqueness which uses matrices with prescribed row and column sums and the dominance order of partitions, see Theorem 1⬘. Our proof uses some identities involving characters of the symmetric group. In a future paper we will study some connections between sets of uniqueness and the Kronecker product of irreducible characters of the symmetric group. Before we state our main theorem we fix some notation. A composition of a positive integer n is a vector s Ž 1 , . . . , p . of non-negative integers whose coordinates sum n. A partition of n is a composition of n such that 1 G 2 G ⭈⭈⭈ G p ) 0. Given two compositions s Ž 1 , . . . , p ., s Ž 1 , . . . , q . of the same number n we say that dominates , in d , if Ý kis1 i G Ý kis1 i for all 1 F k F min p, q4 . If y d symbols y and / , we write d . The dominance order is also called majorization. If is a partition, its conjugate ⬘ is defined by Xi [ < j g w p x < j G i4<. Let s Ž 1 , . . . , p ., s Ž 1 , . . . , q ., s Ž 1 , . . . , r . be fixed compositions of an integer n. We denote by m*Ž , , . the number of subsets S ˜, of B Ž p, q, r . such that S s , S s , and S s . Observe that if resp. ˜ , resp. ˜ is obtained from , resp. , resp. by permutation of ˜, coordinates, then m*Ž ˜ , ˜ . s m*Ž , , .. Also note that if ˆ, resp. ˆ, resp. ˆ is obtained from , resp. , resp. by deleting zeros, then ˆ, m*Ž ˆ , ˆ . s m*Ž , , .. Therefore we may assume without loss of generality that , , are partitions of n. We denote by M Ž , . the set of matrices A with non-negative integer coefficients of size p = q such that its ith row sums i and its jth column sums j . If A g M Ž , . we denote by Ž A. the partition of n obtained from A by ordering its entries decreasingly. We say that A is minimal in M Ž , . if Ž A. is minimal in the set Ž B . < B g M Ž , .4 with the dominance order. Finally, recall that A is called a plane partition if its rows and columns are weakly decreasing. In this note we prove the following characterization for sets of uniqueness in dimension 3, which is similar in spirit to Theorem 2.3 Žsee the comment after the theorem.. THEOREM 1.
Let , , be partitions of n. Then
Ži. m*Ž , , . s 1 if and only if there exists exactly one matrix A g M Ž , . with Ž A. s ⬘ and A is minimal in M Ž , ..
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TORRES-CHAZARO AND VALLEJO ´
Žii. If m*Ž , , . s 1, then the unique matrix A g M Ž , . with Ž A. s ⬘ is a plane partition. Plane partitions correspond to geometric objects we call pyramids. A set S : B Ž p, q, r . is called a pyramid if for all Ž a, b, c . g S and for all Ž x, y, z . g B Ž p, q, r . the conditions x F a, y F b and z F c imply Ž x, y, z . g S. There is a simple well-known one-to-one correspondence between pyramids S : B Ž p, q, r . and plane partitions A s Ž a i j . of size p = q such that 0 F a i j F r, see w7, Sect. 423x. For this reason a pyramid is also called the graph or the diagram of its associated plane partition. The correspondence is given by S ¬ A S s Ž a i j ., where a i j [ < k g w r x < Ž i, j, k . g S 4<; its inverse is A ¬ SA , where SA [ Ž i, j, k . g B Ž p, q, r . < 1 F k F a i j 4 . This correspondence has the following useful property: If S is a pyramid, then A S g M Ž S , S . and Ž A S . s SX . Conversely, if A g M Ž , . is a plane partition and Ž A. s ⬘, then SA has slice vectors , , and . Using this property we reformulate Theorem 1 as follows: THEOREM 1⬘. Let S : B Ž p, q, r . and suppose that its slice ¨ ectors S , S , and S are partitions of the cardinality of S. Then Ži. If S is a set of uniqueness, then S is a pyramid, and its associated plane partition A S is the only matrix in M Ž S , S . with Ž A S . s SX . Moreo¨ er A S is minimal in M Ž S , S .. Žii. Suppose S is a pyramid and let A S denote its associated plane partition. If A S is the only matrix in M Ž S , S . satisfying Ž A S . s SX and if A S is minimal in M Ž S , S ., then S is a set of uniqueness. EXAMPLES. Ž1. For 1 F a - p, 1 F b - q, 1 F c - r we define the hook set S s H Ž a, b, c . as the union of the boxes B Ž a q 1, 1, 1., B Ž1, b q 1, 1., and B Ž1, 1, c q 1.. S is a pyramid with slice vectors S s Ž b q c q 1, 1a ., S s Ž a q c q 1, 1b ., and S s Ž a q b q 1, 1c .. Its corresponding plane partition is cq1 1 .. AS s . 1
1 0 .. . 0
⭈⭈⭈ ⭈⭈⭈ .. . ⭈⭈⭈
1 0 .. . . 0
It is easy to prove that A S satisfies the conditions in Theorem 1⬘Žii.. Therefore S is a set of uniqueness.
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447
Ž2. Let S be the pyramid corresponding to the plane partition 3 As 2 2
3 1 0
1 1 . 0
Then S s S s S s Ž7, 4, 2.. Since the transpose AT is in M Ž S , S . and Ž A. s Ž AT ., then S is not a set of uniqueness. This is the minimal example of a pyramid which is not a set of uniqueness, see w12, 3.3 and 5.1x. It can be checked either by computer or by tedious hand calculations that A is minimal in M Ž S , S .. The paper is organized as follows: Section 2 contains several known results about characters of the symmetric group and matrices. In Section 3 we prove Theorem 1. Finally in Section 4 we give the complete list of all minimal matrices of size 2 = 2.
2. MATRICES AND CHARACTERS There is an elementary one-to-one correspondence between the subsets S of B Ž p, q, r . and 3-dimensional Ž0, 1.-matrices of size p = q = r given by S ¬ MS s Ž a i jk ., where a i jk s
½
1, 0,
if Ž i , j, k . g S; otherwise.
The slice ¨ ectors of a 3-dimensional Ž0, 1.-matrix M s Ž a i jk . of size p = q = r are defined similarly as in Ž1.: M [ Ž 1 , . . . , p ., M [ Ž 1 , . . . , q ., and M [ Ž 1 , . . . , r . where i s Ý j, k a i jk , j s Ý i, k a i jk , and k s Ý i, j a i jk . Clearly for every S : B Ž p, q, r . one has S s M S , S s M S , and S s M S . Therefore for any three partitions , , of n, m*Ž , , . is equal to the number of 3-dimensional Ž0, 1.-matrices M such that M s , M s , M s . The proof of Theorem 1 will be based on a formula due to Snapper which expresses m*Ž , , . in terms of an inner product of certain characters of the symmetric group. We review this and other similar results below. Let s Ž 1 , . . . , p ., s Ž 1 , . . . , q ., and s Ž 1 , . . . , r . be partitions of a positive integer n, and denote by m*Ž , . the number of Ž0, 1.-matrices A of size p = q whose ith row sums i and whose jth column sums j . Consider the following characters of the symmetric group SŽ n.: Let be the permutation character associated to the partition and let ␣ n be the alternating character of SŽ n., see w2, 5, 10x. Finally let ² ⭈ , ⭈ : denote the usual inner product of characters. We will need the following results.
TORRES-CHAZARO AND VALLEJO ´
448
2.1. THEOREM w4, 8, 9x.
e ⬘. m*Ž , . ) 0 m y
2.2. THEOREM w1, Theorem 16; 2, Theorem 2; 5, 1.3x. ² , ␣ n :. 2.3. THEOREM w9, p. 62; 11, Theorem 8.1; 6, p. 83x. s ⬘.
m*Ž , . s
m*Ž , . s 1 m
This result has a geometric reformulation w3, Theorem 3x: Let S : B Ž p, q . s w p x = w q x. A bad rectangle for S, see w3, p. 150x, consists of two distinct points x 1, x 2 in S, and two distinct points z 1, z 2 in B Ž p, q . R S such that Ži. they are the vertices of a rectangle with sides parallel to the coordinate axes, and Žii. x 1 and x 2 Žand hence z 1 and z 2 . are diagonally opposite in the rectangle. Then S is a set of uniqueness if and only if S has no bad rectangle. If S and S are partitions of the cardinality of S, and not merely compositions, then the absence of bad rectangles is equivalent to S s XS . So Theorem 2.3 gives an algebraic characterization of uniqueness in dimension 2. Our Theorem 1Ži. gives a characterization of uniqueness for dimension 3 in the same spirit. Coleman’s result holds for multidimensional matrices, in particular: 2.4. THEOREM w11, Theorem 7.1x.
m*Ž , , . s ² , ␣ n :.
2.5. THEOREM w2, Proposition 2; 5, 2.9.16x.
s Ý A g M Ž , . Ž A..
3. PROOF OF THEOREM 1 Ži. It follows from Ž2.4., Ž2.5., and Ž2.2. that m* Ž , , . s
Ý
m* Ž Ž A . , . .
AgM Ž , .
Suppose m*Ž , , . s 1. Let A g M Ž , . be the only matrix such that m*Ž Ž A., . ) 0. Then by Ž2.3., Ž A. s ⬘. If B g M Ž , . is different e r ⬘ s Ž A.. Therefrom A, then m*Ž Ž B ., . s 0 and by Ž2.1., Ž B . y fore A is minimal in M Ž , .. The converse is similar. Žii. Let m*Ž , , . s 1, and denote by A the only matrix in M Ž , . with Ž A. s ⬘. Suppose A is not a plane partition. Since rows and columns play symmetric roles we may assume that row i is not decreasing. Let k - l be such that a i k - a il . Since k G l , there exists j / i such that a jk ) a jl . Let B be obtained from A by replacing the submatrix ai k q 1
ail y 1
a jk y 1
a jl q 1
. Clearly B g M Ž , .. We define an auxiliary matrix B˜ by replacing in A the submatrix w a i k , a il x by the e Ž B˜. y e Ž A.. Since submatrix w a i k q 1, a il y 1x. Then clearly Ž B . y aik a jk
ail a jl
by the submatrix
449
SETS OF UNIQUENESS AND MATRICES
m*Ž , , . s 1, it follows from Ži. that Ž B . / Ž A.. This contradicts the minimality of A. Therefore A has to be a plane partition. 4. MINIMAL 2 = 2 MATRICES Let , be partitions of n with two parts. In this section we find all minimal matrices in M Ž , .. Since and play symmetric roles we may assume that 1 G 1 G 2 G 2 ) 0. 4.1. THEOREM. The minimal matrices in M Ž , . are: Ži.
If 1 s 2 a q 1, 2 s 2 b q 1, a G b G 0, 1 s 2 , a bq1
Žii.
aq1 , b
a . b
If 1 ) 2 , 2 s 2 b q 1, b G 0,
1 y b y i y 1 bqiq1 Živ.
a . bq1
If 1 s 2 a, 2 s 2 b, a G b ) 0, 1 s 2 , a b
Žiii.
aq1 b
2 y b q i , byi
0 F i F min b, 12 Ž 1 y 2 y 1 . 4 .
If 1 ) 2 , 2 s 2 b, b ) 0,
1 y b y i bqi
2 y b q i , byi
0 F i F min b, 12 Ž 1 y 2 . 4 .
4.2. COROLLARY. Let S : B Ž2, 2, r . be such that S , S , S are partitions. Then S is a set of uniqueness if and only if S is a pyramid. The rest of the section is devoted to prove the theorem and its corollary. 4.3. LEMMA. Let A, B g M Ž , . be such that A / B and Ž A. s Ž B .. Then 1 s 2 and B is obtained from A by permutation of columns. Proof. Let f s 1 y 2 s 1 y 2 . Then there exist 0 F i, j F 2 such fq i yi fqj yj that A s y i 2 i , B s y j 2 j . If A / B and Ž A. s Ž B ., a 2
2
simple case by case analysis shows that B s
2 y i
fqi
i
2 y i
.
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4.4. LEMMA. Let A g M Ž , .. Ži. If 1 ) 2 and A is minimal in M Ž , ., then A is a plane partition. Žii. If A is a plane partition, then A is minimal in M Ž , .. Proof. Ži. Since 1 ) 2 , by Lemma 4.3 there is no B g M Ž , . with Ž A. s Ž B .. Let s Ž A.⬘. Then it follows from Theorem 1 that m*Ž , , . s 1 and that A is a plane partition. Žii. Let A s ac db be a plane partition. Since 1 G 1 , then a G b G c G d, that is, Ž A. s qi byi Ž a, b, c, d .. If B g M Ž , . and B / A, then either B s ac y , 0-i i dqi ay j
bqj
F c, or B s c q j d y j , 0 - j F d. In the first situation the first part of Ž B . is a q i ) a, therefore Ž B .d Ž A. or both partitions are not comparable. In the second situation the last part of Ž B . is d y j - d, again Ž B .d Ž A. or both partitions are not comparable. Then A is minimal. Proof of Theorem 4.1. Ži. Since < M Ž , .< s 2 b q 2, the matrices come in pairs, Ai s
ayi bqiq1
aqiq1 , byi
Bi s
aqiq1 byi
ayi , bqiq1
0 F i F b, d Ž A 0 .. If a ) b, we extend and Ž A i . s Ž Bi .. If a s b, clearly Ž A i . y the notion of dominance to any pair of vectors of non-negative integers of d Ž a q i q 1, a y i, b q i q 1, b y i . y d Ža the same length. Then Ž A i . y . Ž . Ž . Ž . q 1, a, b q 1, b s A 0 . Part ii is similar to i . Žiii. By Lemma 4.4 the minimal elements of M Ž , . and plane partitions coincide. The matrices in Žiii. satisfying 0 F i F min b, 12 Ž 1 y 2 y 1.4 are precisely the plane partitions. Part Živ. is similar to Žiii.. Proof of Corollary 4.2. One implication is a particular case of Theorem 1⬘. For the other suppose S is a pyramid. Then its associated matrix A S is a plane partition. By Lemma 4.4, A S is minimal. It follows from Lemma 4.3 and Theorem 4.1 that there is no other B g M Ž S , S . with Ž B . s Ž A S .. Finally Theorem 1⬘ implies that S is a set of uniqueness. 5. DISCUSSION The class of sets of uniqueness is difficult to grasp. In general it is not easy to prove that a set is a set of uniqueness, because one has to prove the non-existence of a large number of obstructions to uniqueness: the k-bad configurations, see w3, Theorem 1x. One way to overcome this difficulty is by considering additi¨ e sets. This notion was introduced in w3x where the authors proved that additive sets are sets of uniqueness w3,
SETS OF UNIQUENESS AND MATRICES
451
Theorem 2x. In many cases it is easy to prove additivity, see, for example, w12, Theorem 3.4x. Another approach is by considering pyramids. They admit less k-bad configurations than an arbitrary set, see w12, Proposition 3.2 and Theorem 5.1x. In fact it follows from our Theorem 1⬘ that a set of uniqueness is, up to permutation of slices, a pyramid. So it is enough to give a characterization of uniqueness for pyramids. The same Theorem 1⬘ provides such a characterization. However, given a plane partition A g M Ž , . we do not have methods for deciding whether A is the only matrix in M Ž , . with Ž A. s or whether A is minimal in M Ž , .. These are open problems which seem to be difficult and we propose for further study.
REFERENCES 1. A. J. Coleman, Induced representations with applications to Sn and GLŽ n., in Queen’s Papers in Pure and Applied Mathematics, Vol. 4, Queen’s University, Kingston, Ontario, 1966. 2. P. Doubilet, J. Fox, and G. R. Rota, The elementary theory of the symmetric group, in ‘‘Combinatorics, Representation Theory and Statistical Methods in Groups ŽT. V. Narayama, R. M. Mathsen, and J. G. Williams, eds.., Lecture Notes in Pure and Applied Mathematics, Vol. 57, Decker, New York, 1980. 3. P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, Sets uniquely determined by projections on axes. II. Discrete case, Discrete Math. 91 Ž1991., 149᎐159. 4. D. Gale, A theorem of flows in networks, Pacific J. Math. 7 Ž1957., 1073᎐1082. 5. G. D. James and A. Kerber, The representation theory of the symmetric group, in Encyclopedia of Mathematics and Its Applications, Vol. 16, Addison᎐Wesley, Reading, MA, 1981. 6. T. Y. Lam, Young diagrams, Schur functions, the Gale᎐Ryser theorem and a conjecture of Snapper, J. Pure Appl. Algebra 10 Ž1977., 81᎐94. 7. P. A. MacMahon, ‘‘Combinatorial Analysis,’’ Vol. II, Cambridge Univ. Press, LondonrNew York, 1916; reprint, Chelsea, New York, 1960. 8. H. J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 Ž1957., 371᎐377. 9. H. J. Ryser, Combinatorial mathematics, in Carus Math. Monographs, Vol. 14, Math. Assoc. of America, Wiley, New York, 1963. 10. B. E. Sagan, ‘‘The Symmetric Group,’’ Wadsworth & BrooksrCole, Pacific Grove, CA, 1991. 11. E. Snapper, Group characters and nonnegative integral matrices, J. Algebra 19 Ž1971., 520᎐535. 12. E. Vallejo, Reductions of additive sets, sets of uniqueness and pyramids, Discrete Math. 173 Ž1997., 257᎐267.
JOURNAL OF ALGEBRA ARTICLE NO.
Sets of Uniqueness and Minimal Matrices Adolfo Torres-Chazaro ´ Departamento de Matematicas, Uni¨ ersidad Autonoma Metropolitana, Unidad ´ ´ Iztapalapa, Michoacan 09340 Mexico, D.F., Mexico ´ y Purısima, ´ ´
and Ernesto Vallejo* Instituto de Matematicas, Uni¨ ersidad Nacional Autonoma de Mexico, ´ ´ ´ Area de la In¨ . Cient., 04510 Mexico, D.F., Mexico ´ E-mail: [email protected] Communicated by Georgia Benkart Received May 5, 1997
In this note we give an algebraic characterization of sets of uniqueness in terms of matrices with non-negative integer coefficients and prescribed row and column sums, and of the dominance order of partitions or majorization. Our proof uses some identities involving characters of the symmetric group. As an application we describe all sets of uniqueness contained in the box B Ž2, 2, r ., for all r. 䊚 1998 Academic Press
1. INTRODUCTION For a positive integer m let w m x [ 1, . . . , m4 . Given a subset S of the 3-dimensional box B s B Ž p, q, r . [ w p x = w q x = w r x we consider the cardinalities of its slices parallel to the coordinate planes
i s Ž x1 , x 2 , x 3 . g S x1 s i4 ,
1 F i F p,
j s Ž x1 , x 2 , x 3 . g S x 2 s j4 ,
1 F j F q,
k s Ž x1 , x 2 , x 3 . g S x 3 s k4 ,
1 F k F r.
* Supported by DGAPA UNAM IN-103195 and IN-103397. 444 0021-8693r98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.
Ž 1.
SETS OF UNIQUENESS AND MATRICES
445
The slice ¨ ectors S [ Ž 1 , . . . , p ., S [ Ž 1 , . . . , q ., S [ Ž 1 , . . . , r . are compositions of the cardinality of S. The set S is called a set of uniqueness if it is the only subset of B Ž p, q, r . with slice vectors equal to S , S , S . Sets of uniqueness were introduced in w3x, where a geometric characterization of them was given by the absence of certain configurations in B Ž p, q, r .. Their work can be viewed as an extension of previous work on similar questions in the context of Boolean function theory, switching circuit theory, and game theory Žsee references in w3x.. In this note we give an algebraic characterization of sets of uniqueness which uses matrices with prescribed row and column sums and the dominance order of partitions, see Theorem 1⬘. Our proof uses some identities involving characters of the symmetric group. In a future paper we will study some connections between sets of uniqueness and the Kronecker product of irreducible characters of the symmetric group. Before we state our main theorem we fix some notation. A composition of a positive integer n is a vector s Ž 1 , . . . , p . of non-negative integers whose coordinates sum n. A partition of n is a composition of n such that 1 G 2 G ⭈⭈⭈ G p ) 0. Given two compositions s Ž 1 , . . . , p ., s Ž 1 , . . . , q . of the same number n we say that dominates , in d , if Ý kis1 i G Ý kis1 i for all 1 F k F min p, q4 . If y d symbols y and / , we write d . The dominance order is also called majorization. If is a partition, its conjugate ⬘ is defined by Xi [ < j g w p x < j G i4<. Let s Ž 1 , . . . , p ., s Ž 1 , . . . , q ., s Ž 1 , . . . , r . be fixed compositions of an integer n. We denote by m*Ž , , . the number of subsets S ˜, of B Ž p, q, r . such that S s , S s , and S s . Observe that if resp. ˜ , resp. ˜ is obtained from , resp. , resp. by permutation of ˜, coordinates, then m*Ž ˜ , ˜ . s m*Ž , , .. Also note that if ˆ, resp. ˆ, resp. ˆ is obtained from , resp. , resp. by deleting zeros, then ˆ, m*Ž ˆ , ˆ . s m*Ž , , .. Therefore we may assume without loss of generality that , , are partitions of n. We denote by M Ž , . the set of matrices A with non-negative integer coefficients of size p = q such that its ith row sums i and its jth column sums j . If A g M Ž , . we denote by Ž A. the partition of n obtained from A by ordering its entries decreasingly. We say that A is minimal in M Ž , . if Ž A. is minimal in the set Ž B . < B g M Ž , .4 with the dominance order. Finally, recall that A is called a plane partition if its rows and columns are weakly decreasing. In this note we prove the following characterization for sets of uniqueness in dimension 3, which is similar in spirit to Theorem 2.3 Žsee the comment after the theorem.. THEOREM 1.
Let , , be partitions of n. Then
Ži. m*Ž , , . s 1 if and only if there exists exactly one matrix A g M Ž , . with Ž A. s ⬘ and A is minimal in M Ž , ..
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TORRES-CHAZARO AND VALLEJO ´
Žii. If m*Ž , , . s 1, then the unique matrix A g M Ž , . with Ž A. s ⬘ is a plane partition. Plane partitions correspond to geometric objects we call pyramids. A set S : B Ž p, q, r . is called a pyramid if for all Ž a, b, c . g S and for all Ž x, y, z . g B Ž p, q, r . the conditions x F a, y F b and z F c imply Ž x, y, z . g S. There is a simple well-known one-to-one correspondence between pyramids S : B Ž p, q, r . and plane partitions A s Ž a i j . of size p = q such that 0 F a i j F r, see w7, Sect. 423x. For this reason a pyramid is also called the graph or the diagram of its associated plane partition. The correspondence is given by S ¬ A S s Ž a i j ., where a i j [ < k g w r x < Ž i, j, k . g S 4<; its inverse is A ¬ SA , where SA [ Ž i, j, k . g B Ž p, q, r . < 1 F k F a i j 4 . This correspondence has the following useful property: If S is a pyramid, then A S g M Ž S , S . and Ž A S . s SX . Conversely, if A g M Ž , . is a plane partition and Ž A. s ⬘, then SA has slice vectors , , and . Using this property we reformulate Theorem 1 as follows: THEOREM 1⬘. Let S : B Ž p, q, r . and suppose that its slice ¨ ectors S , S , and S are partitions of the cardinality of S. Then Ži. If S is a set of uniqueness, then S is a pyramid, and its associated plane partition A S is the only matrix in M Ž S , S . with Ž A S . s SX . Moreo¨ er A S is minimal in M Ž S , S .. Žii. Suppose S is a pyramid and let A S denote its associated plane partition. If A S is the only matrix in M Ž S , S . satisfying Ž A S . s SX and if A S is minimal in M Ž S , S ., then S is a set of uniqueness. EXAMPLES. Ž1. For 1 F a - p, 1 F b - q, 1 F c - r we define the hook set S s H Ž a, b, c . as the union of the boxes B Ž a q 1, 1, 1., B Ž1, b q 1, 1., and B Ž1, 1, c q 1.. S is a pyramid with slice vectors S s Ž b q c q 1, 1a ., S s Ž a q c q 1, 1b ., and S s Ž a q b q 1, 1c .. Its corresponding plane partition is cq1 1 .. AS s . 1
1 0 .. . 0
⭈⭈⭈ ⭈⭈⭈ .. . ⭈⭈⭈
1 0 .. . . 0
It is easy to prove that A S satisfies the conditions in Theorem 1⬘Žii.. Therefore S is a set of uniqueness.
SETS OF UNIQUENESS AND MATRICES
447
Ž2. Let S be the pyramid corresponding to the plane partition 3 As 2 2
3 1 0
1 1 . 0
Then S s S s S s Ž7, 4, 2.. Since the transpose AT is in M Ž S , S . and Ž A. s Ž AT ., then S is not a set of uniqueness. This is the minimal example of a pyramid which is not a set of uniqueness, see w12, 3.3 and 5.1x. It can be checked either by computer or by tedious hand calculations that A is minimal in M Ž S , S .. The paper is organized as follows: Section 2 contains several known results about characters of the symmetric group and matrices. In Section 3 we prove Theorem 1. Finally in Section 4 we give the complete list of all minimal matrices of size 2 = 2.
2. MATRICES AND CHARACTERS There is an elementary one-to-one correspondence between the subsets S of B Ž p, q, r . and 3-dimensional Ž0, 1.-matrices of size p = q = r given by S ¬ MS s Ž a i jk ., where a i jk s
½
1, 0,
if Ž i , j, k . g S; otherwise.
The slice ¨ ectors of a 3-dimensional Ž0, 1.-matrix M s Ž a i jk . of size p = q = r are defined similarly as in Ž1.: M [ Ž 1 , . . . , p ., M [ Ž 1 , . . . , q ., and M [ Ž 1 , . . . , r . where i s Ý j, k a i jk , j s Ý i, k a i jk , and k s Ý i, j a i jk . Clearly for every S : B Ž p, q, r . one has S s M S , S s M S , and S s M S . Therefore for any three partitions , , of n, m*Ž , , . is equal to the number of 3-dimensional Ž0, 1.-matrices M such that M s , M s , M s . The proof of Theorem 1 will be based on a formula due to Snapper which expresses m*Ž , , . in terms of an inner product of certain characters of the symmetric group. We review this and other similar results below. Let s Ž 1 , . . . , p ., s Ž 1 , . . . , q ., and s Ž 1 , . . . , r . be partitions of a positive integer n, and denote by m*Ž , . the number of Ž0, 1.-matrices A of size p = q whose ith row sums i and whose jth column sums j . Consider the following characters of the symmetric group SŽ n.: Let be the permutation character associated to the partition and let ␣ n be the alternating character of SŽ n., see w2, 5, 10x. Finally let ² ⭈ , ⭈ : denote the usual inner product of characters. We will need the following results.
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2.1. THEOREM w4, 8, 9x.
e ⬘. m*Ž , . ) 0 m y
2.2. THEOREM w1, Theorem 16; 2, Theorem 2; 5, 1.3x. ² , ␣ n :. 2.3. THEOREM w9, p. 62; 11, Theorem 8.1; 6, p. 83x. s ⬘.
m*Ž , . s
m*Ž , . s 1 m
This result has a geometric reformulation w3, Theorem 3x: Let S : B Ž p, q . s w p x = w q x. A bad rectangle for S, see w3, p. 150x, consists of two distinct points x 1, x 2 in S, and two distinct points z 1, z 2 in B Ž p, q . R S such that Ži. they are the vertices of a rectangle with sides parallel to the coordinate axes, and Žii. x 1 and x 2 Žand hence z 1 and z 2 . are diagonally opposite in the rectangle. Then S is a set of uniqueness if and only if S has no bad rectangle. If S and S are partitions of the cardinality of S, and not merely compositions, then the absence of bad rectangles is equivalent to S s XS . So Theorem 2.3 gives an algebraic characterization of uniqueness in dimension 2. Our Theorem 1Ži. gives a characterization of uniqueness for dimension 3 in the same spirit. Coleman’s result holds for multidimensional matrices, in particular: 2.4. THEOREM w11, Theorem 7.1x.
m*Ž , , . s ² , ␣ n :.
2.5. THEOREM w2, Proposition 2; 5, 2.9.16x.
s Ý A g M Ž , . Ž A..
3. PROOF OF THEOREM 1 Ži. It follows from Ž2.4., Ž2.5., and Ž2.2. that m* Ž , , . s
Ý
m* Ž Ž A . , . .
AgM Ž , .
Suppose m*Ž , , . s 1. Let A g M Ž , . be the only matrix such that m*Ž Ž A., . ) 0. Then by Ž2.3., Ž A. s ⬘. If B g M Ž , . is different e r ⬘ s Ž A.. Therefrom A, then m*Ž Ž B ., . s 0 and by Ž2.1., Ž B . y fore A is minimal in M Ž , .. The converse is similar. Žii. Let m*Ž , , . s 1, and denote by A the only matrix in M Ž , . with Ž A. s ⬘. Suppose A is not a plane partition. Since rows and columns play symmetric roles we may assume that row i is not decreasing. Let k - l be such that a i k - a il . Since k G l , there exists j / i such that a jk ) a jl . Let B be obtained from A by replacing the submatrix ai k q 1
ail y 1
a jk y 1
a jl q 1
. Clearly B g M Ž , .. We define an auxiliary matrix B˜ by replacing in A the submatrix w a i k , a il x by the e Ž B˜. y e Ž A.. Since submatrix w a i k q 1, a il y 1x. Then clearly Ž B . y aik a jk
ail a jl
by the submatrix
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m*Ž , , . s 1, it follows from Ži. that Ž B . / Ž A.. This contradicts the minimality of A. Therefore A has to be a plane partition. 4. MINIMAL 2 = 2 MATRICES Let , be partitions of n with two parts. In this section we find all minimal matrices in M Ž , .. Since and play symmetric roles we may assume that 1 G 1 G 2 G 2 ) 0. 4.1. THEOREM. The minimal matrices in M Ž , . are: Ži.
If 1 s 2 a q 1, 2 s 2 b q 1, a G b G 0, 1 s 2 , a bq1
Žii.
aq1 , b
a . b
If 1 ) 2 , 2 s 2 b q 1, b G 0,
1 y b y i y 1 bqiq1 Živ.
a . bq1
If 1 s 2 a, 2 s 2 b, a G b ) 0, 1 s 2 , a b
Žiii.
aq1 b
2 y b q i , byi
0 F i F min b, 12 Ž 1 y 2 y 1 . 4 .
If 1 ) 2 , 2 s 2 b, b ) 0,
1 y b y i bqi
2 y b q i , byi
0 F i F min b, 12 Ž 1 y 2 . 4 .
4.2. COROLLARY. Let S : B Ž2, 2, r . be such that S , S , S are partitions. Then S is a set of uniqueness if and only if S is a pyramid. The rest of the section is devoted to prove the theorem and its corollary. 4.3. LEMMA. Let A, B g M Ž , . be such that A / B and Ž A. s Ž B .. Then 1 s 2 and B is obtained from A by permutation of columns. Proof. Let f s 1 y 2 s 1 y 2 . Then there exist 0 F i, j F 2 such fq i yi fqj yj that A s y i 2 i , B s y j 2 j . If A / B and Ž A. s Ž B ., a 2
2
simple case by case analysis shows that B s
2 y i
fqi
i
2 y i
.
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4.4. LEMMA. Let A g M Ž , .. Ži. If 1 ) 2 and A is minimal in M Ž , ., then A is a plane partition. Žii. If A is a plane partition, then A is minimal in M Ž , .. Proof. Ži. Since 1 ) 2 , by Lemma 4.3 there is no B g M Ž , . with Ž A. s Ž B .. Let s Ž A.⬘. Then it follows from Theorem 1 that m*Ž , , . s 1 and that A is a plane partition. Žii. Let A s ac db be a plane partition. Since 1 G 1 , then a G b G c G d, that is, Ž A. s qi byi Ž a, b, c, d .. If B g M Ž , . and B / A, then either B s ac y , 0-i i dqi ay j
bqj
F c, or B s c q j d y j , 0 - j F d. In the first situation the first part of Ž B . is a q i ) a, therefore Ž B .d Ž A. or both partitions are not comparable. In the second situation the last part of Ž B . is d y j - d, again Ž B .d Ž A. or both partitions are not comparable. Then A is minimal. Proof of Theorem 4.1. Ži. Since < M Ž , .< s 2 b q 2, the matrices come in pairs, Ai s
ayi bqiq1
aqiq1 , byi
Bi s
aqiq1 byi
ayi , bqiq1
0 F i F b, d Ž A 0 .. If a ) b, we extend and Ž A i . s Ž Bi .. If a s b, clearly Ž A i . y the notion of dominance to any pair of vectors of non-negative integers of d Ž a q i q 1, a y i, b q i q 1, b y i . y d Ža the same length. Then Ž A i . y . Ž . Ž . Ž . q 1, a, b q 1, b s A 0 . Part ii is similar to i . Žiii. By Lemma 4.4 the minimal elements of M Ž , . and plane partitions coincide. The matrices in Žiii. satisfying 0 F i F min b, 12 Ž 1 y 2 y 1.4 are precisely the plane partitions. Part Živ. is similar to Žiii.. Proof of Corollary 4.2. One implication is a particular case of Theorem 1⬘. For the other suppose S is a pyramid. Then its associated matrix A S is a plane partition. By Lemma 4.4, A S is minimal. It follows from Lemma 4.3 and Theorem 4.1 that there is no other B g M Ž S , S . with Ž B . s Ž A S .. Finally Theorem 1⬘ implies that S is a set of uniqueness. 5. DISCUSSION The class of sets of uniqueness is difficult to grasp. In general it is not easy to prove that a set is a set of uniqueness, because one has to prove the non-existence of a large number of obstructions to uniqueness: the k-bad configurations, see w3, Theorem 1x. One way to overcome this difficulty is by considering additi¨ e sets. This notion was introduced in w3x where the authors proved that additive sets are sets of uniqueness w3,
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Theorem 2x. In many cases it is easy to prove additivity, see, for example, w12, Theorem 3.4x. Another approach is by considering pyramids. They admit less k-bad configurations than an arbitrary set, see w12, Proposition 3.2 and Theorem 5.1x. In fact it follows from our Theorem 1⬘ that a set of uniqueness is, up to permutation of slices, a pyramid. So it is enough to give a characterization of uniqueness for pyramids. The same Theorem 1⬘ provides such a characterization. However, given a plane partition A g M Ž , . we do not have methods for deciding whether A is the only matrix in M Ž , . with Ž A. s or whether A is minimal in M Ž , .. These are open problems which seem to be difficult and we propose for further study.
REFERENCES 1. A. J. Coleman, Induced representations with applications to Sn and GLŽ n., in Queen’s Papers in Pure and Applied Mathematics, Vol. 4, Queen’s University, Kingston, Ontario, 1966. 2. P. Doubilet, J. Fox, and G. R. Rota, The elementary theory of the symmetric group, in ‘‘Combinatorics, Representation Theory and Statistical Methods in Groups ŽT. V. Narayama, R. M. Mathsen, and J. G. Williams, eds.., Lecture Notes in Pure and Applied Mathematics, Vol. 57, Decker, New York, 1980. 3. P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, Sets uniquely determined by projections on axes. II. Discrete case, Discrete Math. 91 Ž1991., 149᎐159. 4. D. Gale, A theorem of flows in networks, Pacific J. Math. 7 Ž1957., 1073᎐1082. 5. G. D. James and A. Kerber, The representation theory of the symmetric group, in Encyclopedia of Mathematics and Its Applications, Vol. 16, Addison᎐Wesley, Reading, MA, 1981. 6. T. Y. Lam, Young diagrams, Schur functions, the Gale᎐Ryser theorem and a conjecture of Snapper, J. Pure Appl. Algebra 10 Ž1977., 81᎐94. 7. P. A. MacMahon, ‘‘Combinatorial Analysis,’’ Vol. II, Cambridge Univ. Press, LondonrNew York, 1916; reprint, Chelsea, New York, 1960. 8. H. J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 Ž1957., 371᎐377. 9. H. J. Ryser, Combinatorial mathematics, in Carus Math. Monographs, Vol. 14, Math. Assoc. of America, Wiley, New York, 1963. 10. B. E. Sagan, ‘‘The Symmetric Group,’’ Wadsworth & BrooksrCole, Pacific Grove, CA, 1991. 11. E. Snapper, Group characters and nonnegative integral matrices, J. Algebra 19 Ž1971., 520᎐535. 12. E. Vallejo, Reductions of additive sets, sets of uniqueness and pyramids, Discrete Math. 173 Ž1997., 257᎐267.