Root-Patterns to Algebrising Partitions

Transcription

1 Advances in Pure Maematics, 015, 5, Published Online January 015 in SciRes. Root-Patterns to Algebrising Partitions Rex L. Agacy 4 Brighton Street, Gulliver, Townsville, Australia ragacy@iprimus.com.au Received 6 December 014; revised 7 January 015; accepted January 015 Copyright 015 by auor and Scientific Research Publishing Inc. This work is licensed under e Creative Commons Attribution International License (CC BY). Abstract The study of e confluences of e roots of a given set of polynomials root-pattern problem does not appear to have been considered. We examine e situation, which leads us on to Young tableaux and tableaux representations. This in turn is found to be an aspect of multipartite partitions. We discover, and show, at partitions can be expressed algebraically and can be differentiated and integrated. We show a complete set of bipartite and tripartite partitions, indicating equivalences for e root-pattern problem, for select pairs and triples. Tables enumerating e number of bipartite and tripartite partitions, for small pairs and triples are given in an appendix. Keywords Combinatorics, Partitions, Polynomials, Root-Patterns, Tableaux 1. Introduction We are interested in e root-patterns or confluences of e roots of a given set of polynomials a topic one may have expected would have been studied in dep in e 19 century. However, apart from results on Resultants etc., ere does not appear to have been much furer development. Motivation for consideration here arises from General Relativity where e classification of e Lanczos-Zund (3,1) spinor involves various confluences of e roots of two cubics [1]. This in turn relates to e important aspect of Invariants of e spinor much like e confluences of roots of a quartic form e various algebraic types for e Weyl spinor (Petrov classification) and are linked to its Invariants. It is shown here how e root-patterns problem becomes a problem in partitions. In is context, from bipartite partitions, an application to derivations of spinor factorizations in General Relativity has been made []. For example, given a quadratic and two cubics, a root-pattern may be indicated by ab, aad, bce where a, b are e roots of e quadratic and a, a, b and b, c, e are e roots of each of e cubics. Three observations may be made. Firstly, e order of e polynomials is immaterial and so e root-pattern is also aad, ab, bce, or bce, aad, ab etc. Secondly, alough we have written e roots of each polynomial in ascending letter order, e confluences of e roots or root-pattern is unchanged if we rearrange e letter order How to cite is paper: Agacy, R.L. (015) Root-Patterns to Algebrising Partitions. Advances in Pure Maematics, 5,

2 for each polynomial s roots. Then e last root-pattern is also ebc, ada, ab. Thirdly, for any root-pattern, we can replace any letter by any oer, unused, letter as e actual value of e root is unimportant. This en allows an interchange of letters, e.g., bce, aad, ab wi e interchange a b becomes ace, bbd, ba. In oer words, any permutation of letters is allowed. From an initial set S, a collection of elements of S where elements can be repeated, and shown juxtaposed, is just a list. A list wi r elements is an r-element list and is e degree of e corresponding polynomial whose roots are e elements of e list or tuple. We define a root-pattern as just a collection of lists. Thus, e root-pattern ab, aad, bce is e collection of lists ab and aad and bce where e initial set is { abcde,,,, }. There are ree elements in is root-pattern, a -element list or pair and two 3-element lists. The number of lists in e collection is e number of polynomials. The number of components in a list is e degree of e polynomial. If we take two polynomials, we refer to e binary case, for ree polynomials, e ternary case etc. The ree observations we made earlier may now be formalized as e following rules. 1) Any two polynomials can be interchanged. This translates to any two lists in a root-pattern can be interchanged. ) Any rearrangement of e ordering of roots of a polynomial translates as a rearrangement of e ordering of e roots in e corresponding list for at polynomial in e root-pattern. 3) Any permutation of e letters (roots) in e set of polynomials translates to a permutation of e letters in e root pattern. It is e commonality or confluence of roots in e various polynomials at we are interested in. We define two root-patterns A and B to be equivalent if B can be obtained from A by any of e ree rules; oerwise ey are inequivalent. For e ternary case, e root patterns ab, aad, bce and bc, bvb, xcy are equivalent but neier are equivalent to aad, bb, acc. For a given set of polynomials e various root-patterns, at is e combinations of e various roots, which include common roots, is of interest to us. What we would like to obtain is e enumeration and e consequent collection of (inequivalent) root-patterns of several polynomials. More specifically, given m 1 linear forms, m quadratics,, m n n-ics, enumerate and determine e set of inequivalent root-patterns. This is e root-pattern problem. Let us first consider a simple example. Two Quadratics The roots of two quadratics (two lists of pairs, m = ) are displayed in e form wx, yz where e letters refer to e roots of e two polynomials. Besides using letters we also use corresponding numbers. aa, aa = 11, 11 all roots equal. aa, ab = 11, 1 roots of first are equal, which are common to one root of second. aa, bb = 11, roots of first are equal, but different to equal roots of second. aa, bc = 11, 3 roots of first are equal, but distinct to each root of second. ab, ab = 1, 1 bo distinct roots of first are duplicated in second. ab, ac = 1, 13 roots of first and second have a common root, wi each of e oers distinct. ab, cd = 1, 34 all roots of bo quadratics are different. There are 7 inequivalent cases which can be written as rows = = 3 = 4 = 5 = 6 = 7 = Note, for example, at a case written as ab, cc (1, 33) is essentially e same as cc, ab (33, 1) since e order of e quadratics is immaterial. Then too cc, ab (33, 1) is also e same as case 4 aa, bc (11, 3), since e letters (and numbers) used do not matter.. Young Tableaux It is seen at e roots of e two quadratics, displayed as rows, are instances of Young tableaux. We can always represent e roots of a set of polynomials as Young tableaux. For a polynomial of highest degree, say λ 1, we 3

3 place its roots in a first or top row, and moving downwards to a second row wi polynomials of e same or lesser degrees λ, λ 3 and so on, from top to bottom. The number of entries in row i is λ i and is e leng of e row. Thus for a tableaux of n rows, λ1 λn. The root-patterns of any number of polynomials can be displayed as Young tableaux. As e order of roots of a polynomial in any row is immaterial we will take it at e numbers in each row of a Young tableau are always arranged in weakly increasing order. A question arises as to wheer every tableau can be ordered so at every row and column is weakly increasing. This is in fact not so. For any permutation of S3 = { e, ( 1 ),( 13 ),( 3 ),( 13 ),( 13) } and row interchanges, e tableau (wi weakly increasing rows) cannot be displayed as one wi weakly increasing columns, also allowing for e interchange of any rows..1. Tableau Representation We will take it at e numbers used in any tableau of n rows will be consecutive, 1,,,q (e largest number in e tableau). Should any tableau contain q non-consecutive numbers we can always replace em so at we have consecutive numbering 1,,, q. We now construct a different numbering on a tableau T. For a given T we form n-tuples or lists and ere will be at most q of em. We will write e n-tuples or lists in sequence which we call e tableau representation. The i entry of e j tuple will be e number of times e number j appears in e i row. Thus for e tableau T wi n = 3 rows and q = T = count e number of times 1 appears in e first row of e tableau, next e number of times 1 appears in e second row and so on, written as 110, e first 3-tuple or list. Then count e number of times appears in e first row, next e number of times appears in e second row and so on, to get 00 etc., so at finally we construct e tableau representation 110, 00, 10, 100 of T. From is representation, e tableau can be reconstructed as follows. 1) The sum of e first components of each triple tells us e leng of e first row; similarly for e second and ird. Thus for T we have a 3-rowed tableau wi row lengs λ1, λ, λ 3 = 3,3,. ) Since ere are four triples ere will be 4 different numbers 1,, 3, 4 used. 3) The first component of each 3-tuple tells us at e first row contains one 1, one 3 and one 4, so e first row is 134. Similarly e second row is 133 and e ird row is. Putting ese togeer, one row under anoer, in order, recovers T. In e tableau representation of a set of polynomials, e a tuple refers to e i row of e tableau. The 7 tableaux for two quadratics, namely j tuple refers to e number j, e = = 3 = 4 = 5 = 6 = 7 = i number in have e 7 tableaux representations 1,01 0,0 0,01,01 11,11 11,10,01 10,10,01,01... Nil-Addition and e Algebraic Representations of Partitions Partitions may be represented in two ways: for example e partitions of e number (4) are: 4, 3 + 1, +, + 33

4 , and a second way as: This latter notation is manifested by extracting, from e following products, e terms of degree 4 so at we have ( 1 + x x.1 + x x )( 1 + x 1. + x. + x 3. + )( 1 + x x.3 + )( 1 + x ) x x x x x x x where e exponents are e partitions ,3 1 1 which is e second notation. The general rule for partitions of a single variable is given by a generating function [3]. Let H = { h1, h, h3, } be a set of positive intergers, usually { 1,,3, }. Then m H m 1 m m 3m ( 1 x ) = ( 1+ x + x + x + ) m H h1 h1 3h1 h h 3h ( 1 x x x )( 1 x x x ) = = a1, a, a3 0 x ah 11+ ah+ ah 3 3+ a1 a a3 where e exponent of x is e partition h1 h h3 This is seen in e above parttioning of e number (4). For bipartite partitions e generating function for a set of pairs H = { hk i j} of positive integers, usually 1,1, 1,, 1,3,,,1,,, is { } ( mn, ) H. m n ( 1 x y ) h k Expanding as a power series in x and y, e coefficient of x y is e number of bipartite partitions of ( hk, ). The process can be computerized and short tables of bipartite and tripartite partitions are displayed in e Appendix. We will use e first notation, however, and consider bipartite partitions here. The pair (,1) has e following 4 (bi)partitions: 1 1 0,01 11,10 10,10,01. We have used a delimiting comma here, raer an e usual summation sign, since we will now use e summation sign to express e partitions as a sum 1+ 0, , ,10, 01. Any (bipartite) partition is a tuple/list of parts: us, e partition 11, 10 has parts 11 and 10. Each part is comprised of components; e part 10 has components 1 and 0. Partitions are en shown as eir tableau representations. Thus we can talk of a partition or a tableau representation of a tableau, and construct a tableau at represents e partition 1. The + symbol used here needs more definition. Whilst it will be legitimate to write 0, ,10 ere will be no point in writing 11, ,10 = ( 11,10) since we are only interested in e single partition, not multiples of it. We avoid such multiples of partitions by adopting e rule for addition of a partition a+ a = a. We refer to e rule as e nil-addition of a partition. The root-patterns of two quadratics, written (,) can en be shown as e sum of e 7 tableau representations, so at we now write Note at e tableau representations (, ) = + 1, 01+ 0, 0 + 0, 01, ,11+ 11,10, ,10, 01, ,01 = and ,01,01 = are equivalent respectively to 3 1 1,10 = 11 1 Note at e partitions and tableau representations of e number (4) are: 4 = 1111; = 111; + = 11; = 113; =

5 1 and 10,10, 0 =, e first by interchange of rows; e second, by interchanging e numbers 1 and 3 and en 33 interchanging rows. The root patterns are equivalent for ese two. Including e latter two equivalent tableaux representations to e 7 above, we have (, ) = + 1, 01+ 0, 0 + 0, 01, 01+ 1, ,11+ 11,10, ,10, ,10, 01, 01. This is exactly e expression of e pair (,) into its 9 bipartite partitions. The 7 tableau representations (or corresponding partitions) for e roots of two quadratics are a subset of e full set of tableau representations of all partitions of (,). It is convenient to call e sum of all partitions of a given number, pair, triple etc., its partition representation. Algebraic Representation Parts and hence partitions of a number, pair of numbers etc., can be represented algebraically. For e bipartite case let x represent e tuple 10 and y e tuple 01. Then put x = 0, y = 03 and x y = 45 etc. Such representations of parts or tableau representations we refer to as algebraic representations. The representation by monomials transfers partitions to algebraic notation obeying rules we have yet to set up Let us denote e concatenation of two tuples by a concatenation (circle) symbol ( ), as a concatenating operator, so at 0,01 is written 0 01 and 10,10,01 is This notation will apply to all concatenation of tuples. Besides e concatenation symbol, we introduce an extension symbol ( ) as an extension (or integrating) operator and develop some rules for circle and extension multiplication. Let a, b, c, d be monomials. We define e following laws 11 = 1 (1) a 1 = a () a b = b a (3) a b c d = ac bd + ad bc + ac b d + ad b c + bc a d + bd a c + a b c d (4) ( a b) ( c d + e f ) = ( a b) ( c d) + ( a b) ( e f ) (5) where ac etc., is e ordinary (dot ) multiplication of two monomials. It is easily seen at ( a b) ( c d) = ( c d) ( a b), so at bo and are commutative. Rules (), (3) and (4) allow us to simplify some of e calculations at are used, by employing simplified rules. Thus wi b = 1 in rule 4 we have a 1 c d = ac d + ad c + ac 1 d + ad 1 c + c a d + d a c + a 1 c d = ac d + ad c + ac d + ad c + a c d = ac d + ad c + a c d using nil-addition. Most often we will erefore use rule 4 d = in 4, we have. a c d = ac d + ad c+ a c d (4 ) Then wi 1 a c 1 = a c = ac 1+ a c+ a c 1 which, also wi nil-addition, gives a simple rule 5 which we will very frequently employ a c = ac + a c. (5 ) The term ac is a monomial and e term a c is a concatenated term. It is clear at a c = c a. Rule 5 is a distributive law, and for simplified versions we have In fact e delimiting comma (,) denotes e concatenation but we use e circle symbol as it is maematically more readable and a reminder at a new process is involved. 35

6 ( ) ( ) ( ) 6. a c d + e f = a c d + a e f. Collating e rules at will be frequently used we have 11 = 1 (1) a 1 = a () a b = b a (3) a c d = ac d + ad c+ a c d (4 ) a c = ac + a c (5 ). a c d + e f = a c d + a e f (6 ) Note at (5 ) is derivable from (4 ) and at e rhs of a c consists of a monomial part ac and a concatenated part a c. It is also easily seen at a ( b c) = ( a b) c. In ese shortened versions of e original laws it may be convenient to use e terminology of an integating operator instead of at of e extension operator, using e symbol i raer an. Wi is notation we rewrite laws (4 ), (5 ) and (6 ) as a i c d = ac d + ad c+ a c d (4 ) ia ( c) = ac + a c (5 ) ( + ) = +. i c d e f i c d i e f (6 ) a a a Let a = y, b = x y en rule (5 ) gives a b = y x y = x y + x y y. In terms of tuples where x = 10, y = 01 e monomial expression a b = 01 1 = + 1,01, where we see at e (dot) product y x y = x y or 01 1 = is e mere addition of e components 01 and 1. The left table below shows an example of e use of e extension operator employed algebraically, wi e rhs consisting of monomials, including concatenations of em. The right table shows e interpretation of ese as tableau representations. 1) y x = x y 01 0 = 0,01 ) y x y = x y y = 0,01,01 ( + ) = ( + ) 3) i y x = y x = yx + y x 01 0 = 1+ 0,01 4) i y xy x y y xy x y x y y x y yx y x y y x y = ( ) = ( ) 5) i y x y y x y = + + x y y x y x y y ( + ) = ( + ) y xy y ( x y) 6) i y xy x y y xy x y = + = xy y xy x y y x y x y y , 01 = + 01, 1+ 1,01+ 0,0 + 01, 0, , 01 = 1,01+ 0,0 + 0,01,01 ( ) , 01 = = ,11+ 1, 01+ 0, 0 + 0, 01, 01. Example 1. Suppose we wish to obtain e tableau representation of (,) associated wi e roots of two quadratics. It consists of e 9 partitions (, ) = + 1, 01+ 0, 0 + 0, 01, 01+ 1, ,11+ 11,10, ,10, ,10, 01,

7 We can construct e (,) partition representation from e algebraic representation of (,1) The algebraic,1 = 1+ 0, , ,10, 01 is representation of,1 = x y+ x y+ xy x+ x x y. To get e (,) algebraic representation from e (,1) algebraic representation we multiply it by 01, at is by e monomial y, using e extension operator. Taking each of e 4 terms separately we get, making simplifications and writing terms wi x = 10 preceding y = 01 to reorder monomials in circle products, y x y x y y x y x y x y y = + = + = + + y x y yx y x y y x y x y y x y x y y = + + ( ) = + + = + + y xy x xy x xy xy y xy x xy x xy xy xy x y y x x y = xy x y+ x xy y++ x x y + x x y y = xy x y+ x x y + x x y y. Adding ese up we have e algebraic representation of (,) (, ) = 01 (,1) = y ( x y+ x y+ xy x+ x x y) x y x y y x y y x y x y y xy x xy xy xy x y = xy x y+ x x y + x x y y = x y + x y y + x y + x y y + xy x + xy xy + 3 xy x y + x x y + x x y y. Ignoring e irrelevant numerical coefficients (nil-addition) in is 9 term algebraic representation of (,) we have, = x y + x y y + x y + x y y + xy x + xy xy + xy x y + x x y + x x y y. In terms of e partition representation is is (, ) = + 1, 01+ 0, 0 + 0, 01, 01+ 1, ,11+ 11,10, ,10, ,10, 01, 01. The actual tableau corresponding to each term in ese expressions is easily constructed. In e algebraic representation, e terms x y y and xy x are equivalent (interchange x and y ), as also e terms x y y and x x y. The tableau representation is e equivalence of pairs 1, 01 and 1, 10 and pairs 0, 01, 01 and 10, 10, 0. Thus we may consider e (,) partitioning as consisting of 7 inequivalent pairs. We may take is by defining an order (dominance) algebraically on e variables, say, x > y (us ignoring e term xy x ) or 1,01 > 1,10 (ignoring e latter term). Alternatively we could symmetrize e expression and consider e set of symmetrised terms, (, ) = + ( 1, 01+ 1,10) + 0, 0 + ( 0, 01, ,10, 0) + 11,11+ 11,10, ,10, 01, 01 which are 7. So really e problem of finding e number of inequivalent root-patterns of a set of polynomials (two quadratics here) is subsumed as at of determining e set of symmetrized partitions, a subset of all partitions corresponding to all tableau representations of e polynomials. 3. Differentiation of Partitions Partitons now being expressed algebraically, provide an opportunity to introduce differentiation.,,, 37

8 If a and b are monomials in x (ey must be of e form such at m x and n x ) we define a derivative operator m m 1 m 1 = = = ( using nil-addition of ignoring numerical coefficients) 1. d a d x mx x x x ( ) = + ( rentiation). d a b a d b b d a x x x product rule for diffe. In practice we may just differentiate a monomial and ignore any coefficients. The derivative operator can be extended to a partial derivative operator such as d in an obvious way etc. xy Example. The tableau and algebraic partitioning of numbers (3) and (4) is 3 3 = 3+,1+ 1,1,1 = x + x x+ x x x, 4 = 4 + 3,1+, +,1,1+ 1,1,1,1 = x x x x x x x x x x x x Ordinary differentiation of e latter (also using.) gives. x + x x+ x + x x + x x+ x x x+ x x+ x x + x x x which, ignoring coefficients, reduces to x + x x+ x x x which is precisely e partitioning of (3). The tableau and algebraic partitioning of e bivariate partitions (,1) and (,) is,1 = x y+ x y+ xy x+ x x y,, = x y + x y y + x y + x y y + xy x + xy xy + xy x y + x x y + x x y y. Differentiating e latter wi respect to y gives x y + x y + x y 1+ x y + x y 1 + xy x + xy x + x x y + xy x 1+ x x y + x x y 1 which all amounts to, ignoring coefficients, x y+ x y+ xy x+ x x y precisely e partitioning of (,1). The process of differentiation in e example has exhibited e downgrading of partitions from (4) to (3) and from (,) to (,1). The reverse process of integrating or upgrading was performed in Example 1 in deriving e partitioning of (,) from (,1). 4. Display of Partitioning for Low Degree Polynomials The partition representations of low degree polynomials are displayed. Equivalent partitions are superscripted alike. The first-listed partition is e dominant one. Partitions of all different row lengs are necessarily all inequivalent. Partitions wi equal row lengs will have partition equivalents. The algebraic representations can easily be constructed from e tableau representations Binary Root Patterns Two Quadratics (,) There are 9 possible partitions wi 7 being inequivalent. a b a b, = + 1, , 0 + 0, 01, , ,11+ 11,10, ,10, ,10, 01, Cubic and Quadratic (3,) As e polynomials are of different degrees all partitions are inequivalent. There are 16 inequivalent partitions. 3, = , , , 01, 01+,10 + 1,11+ 1,10, 01+ 0,1 + 0,11, 01+ 0,10, 0 + 0,10, 01, 01+ 1,10, ,11, ,10,10, ,10,10, ,10,10, 01, 01. d x 38

9 Two Cubics (3,3) The total number of partitions is 31. The number of inequivalent ones (here) is 1. 3, 3 = , , , 01, , , 0, , 01, 01, 01+,11+,10, 01+ 1,1 + 1,11, 01+ 1,10, 0 + 1,10, 01, 01+ 0,11, 0 + 0,11, 01, 01+ 0,10, 0, 01+ 0,10, 01, 01, ,11,11+ 11,11,10, ,10,10, 01, ,10,10, 01, 01, 01. Oer partitions, equivalent to some listed here, are: 3,10 3,10 and 10,1,10, 01 1,10, 01, 01, and 10,11, 0,10 0,11, 01, 01 etc. 4.. Ternary Root-Patterns A Cubic and Two Linear Forms (3,1,1) The total number of partitions is 1. The number of inequivalent ones is 17. 3,1,1 = , , , , 010, , , ,100, 001 a a b c b c + 01, ,100, , ,110, ,100, ,100, 010, 001 d d + 111,100, ,101, ,100,100, ,100,100, ,100,100, ,100,100, 010, 001. It is easily seen at a partition is equivalent to a given partition if e i and j entries of each part of e partition are interchanged corresponding to row interchanges in e tableau view of e partitions. More generally, any permutation of e numbers of a permutation gives an equivalent partition. Interchange of e parts of a partition produces an equivalent partition Two Quadratics and a Linear Form (,,1) The total number of partitions is 6. The number of inequivalent ones is 0. a b c d,,1 = 1+ 0, , , , 010, , , 010, , , 00, , 011, , 010, 010, , , ,100, 001 e a b c + 111, ,100, ,110, ,101, ,100, ,100, 010, 001 e d f f + 101,100, ,100, ,100, 010, ,100, 011, ,100, 010, 010, Three Quadratics (,,) The total number of partitions is 66. The number of inequivalent ones is 51.,, = + 1, , , 001, , , , 010, , 01 a b a c d e f g h + 10, 011, , 010, , 010, 001, , , 010, , , 00, , 011, , 010, 010, , , 01, 001 i j + 00, 00, , 00, 001, , 01, , 011, , 011, 010, 001 k + 00,010,010, ,010,010,001, , , ,100, ,10 f g h c d + 10,101, ,100, ,100, 001, , ,100, ,111 m + 111,110, ,100, ,100, 010, , 110, ,110, 001, ,10, ,101, ,101, 010, ,100, ,100, 011, 001 n e j + 110,100, 010, 001, , ,110, ,100, ,100, 010,010 i f + 101,101, ,101, 010, ,100, ,100, 00, ,100, 011, 010 b k + 101,100, 010, 010, ,100, ,100, 01, ,100, 00, 00 l m n + 100,100, 00, 001, ,100, 01, ,100, 011, ,100, 011, 010, ,100, 010, 010, ,100, 010, 010, 001, 001. l d e 39

10 References [1] Agacy, R.L. and Briggs, J.R. (1994) Algebraic Classification of e Lanczos Tensor by Means of Its (3,1) Spinor Equivalent. Tensor, 55, [] Agacy, R.L. (00) Spinor Factorizations for Relativity. General Relativity and Gravitation, 34, [3] Andrews, G.E. (1984) The Theory of Partitions. Cambridge University Press, Cambridge. Appendix 1. Bipartite Partitions The table shows e number of bipartite partitions for (, ) in a power series expansion of 0 mn, 9 r s rs for 0 r, s 9, which is e coefficient of xx 1 m n ( x ) 1 1 x 1. s = r = r = r = r = r = r = r = r = r = r = ,715,65. Tripartite Partitions The tables show e number of tripartite partitions for (,, ) xxx in a power series expansion of r s t mn,, p 4 rst for 0 r, s, t 4, which is e coefficient of ( m n p x ) 1 1 xx

11 t = r = 0 s = r = 0 s = r = 0 s = r = 0 s = r = 0 s = t = r = 1 s = r = 1 s = r = 1 s = r = 1 s = r = 1 s = t = r = s = r = s = r = s = r = s = r = s = t = r = 3 s = r = 3 s = r = 3 s = r = 3 s = r = 3 s = t = r = 4 s = r = 4 s = r = 4 s = r = 4 s = r = 4 s =

12