[31D.E. Littlewood, The Theory of Group Characters, 2nd ed.

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1 Computer Physics Communications 28 (1982) North-Holland Publishing Company COMPUTATION OF OUTER PRODUCTS OF SCHUR FUNCTIONS Omer EGECIOGLU Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA Received 17 May 1982 PROGRAM SUMMARY Title ofprogram: Schur Method of solution A new backtracking algorithm [1] is implemented to generate Catalogue number: AAMJ the partitions that appear in the expansion of a product of Schur functions. Program obtainable from: CPC Program Library, Queen s University of Belfast, N. Ireland (see application form in this issue) Restrictions on the complexity of the problem The size of the problem that can be handled by the implemen- Computers upon which the program is operable: Various mini tation is restricted by the total number of parts of the input and microcomputers supporting the UCSD Pascal system partitions (no more than 255). Computer: PDP-l 1/780; Installation: UCSD Operating system: UCSD Pascal system Programming language used: UCSD Pascal High speed storage required: 1900 words Number ofbits in a word: 16 Overlay structure: none Typical running time (52){322) 2s. (31)(22)(21){12) 8 s. (Time indicated includes the disk-write time: I and 2 s, respectively.) Unusualfeatures of the program The partitions generated by the algorithm are maintained as nodes of a balanced binary tree to minimize list insertion time. Peripherals used: console, disk Number of cards in combined program and test deck: 498 References [1] J. Remmel and R. Whitney, Multiplication of Schur Func- tions (to appear). [2] D.E. Littlewood and A.R. Richardson, Phil. Trans. A 233 Keywords: Schur function, outer product, irreducible represen- (1934) 99. [31D.E. Littlewood, The Theory of Group Characters, 2nd ed. tation, partition, Ferrers diagram, standard tableau, skewtableau, backtracking, depth-first order, balanced tree (Oxford Univ. Press, London, 1950) p. 94. Nature ofphysical problem To express the (outer) product of an arbitrary number of Schur functions as a linear combination of Schur functions /82/ /$ North-Holland

2 E~ecio1lu/ Outer products of Schur functions LONG WRITE-UP 1. Infroduction Efi The product of two Schur functions (A ) and (A 2) of degrees m and n, respectively, can be expressed Fig. 1. Representation of the partition A = (4 3 2). as a nonnegative integral linear combination of Schur functions (ps) of degree m + n: consists of A. squares, = 1, 2,..., k. For example (A ){A2) = ~ g~a~{~) (l) the partition A = (432) is represented as shown in fig. 1. This is also referred to as the shape of A. The correspondence [2] between the construction If E~ =,X, = n then we denote this by A H n and of the outer product A x A2 of two irreducible put XI = n. ST(A) is the collection of all standard representations A and A2 of the symmetric group tableaux of shape A and and the (outer) product of the Schur functions determined by these partitions gives the multiplici- ST( n): = XHn U ST( A). ties of the irreducible constituents of the representation A x A2, once the expansion (1) is known. If r E ST(A) then we set P( r) = A. Similarly, the coefficient g~l~2 ~ in the expan- If A, A2 are two partitions, the skew-shape F( A, A2) is the plane figure (fig. 2). The lexicographic filling LF( A, A2) of a skew- {A )(A2)... (Ar) = ~g~i~2 {~) (2) shape F(A, A2) is obtained by filling the cells of the underlying skew-shape with the integers 1, of the Schur functions (A ), (A2),...,(X ) gives the 2,..., IA I + 1A21 (in that order) starting from the multiplicity of the corresponding irreducible repre- upper left cell and proceeding row-wise down the sentation p~in the analysis of A x A2 x... x A~. figure. These definitions clearly extend to an arbi- In ref. [2], a combinatorial rule was given to trary number of partitions A, A2,..., A~.If compute the coefficients g~i~2 that occur in (l) {A )(A2)... (At) = ~ ~ (See also ref. [3]). We implement a new method obtained in ref. [1], which makes it possible to multiply an arbi- where N = IA I + 1A A ~then we can corntrary number of Schur functions directly, and thus pute the coefficients g~i~2 ~ by the following obtain the coefficients g~i~2 that appear in (2) version of the result of ref. [1]: without excessive computation. As a surprising by-product of the algorithm, the Theorem. Let LF( A, A2,..., A~) be the lexicodegrees n ~of the irreducible representations of the graphic filling of the skew-shape F( A, A2,..., At) symmetric group can also be computed without much effort in view of the identity ~ n~(a). 2. Terminology and the method A partition A = (A, A 2... A~)is pictori- Fig. 2. Plane figure of the skew-shape F(A, A2). ally represented via its Ferrers diagram by taking k left-justified rows of squares where the ith row Fig. 3. LF(A, A2) with Ai = (2 1) and A2 = (12).

3 III 0. E~ecio~lu/ Outer products of Schurfunctions Computer implementation I _ ~ 1~13I4I 3.1. An overview The program generates and maintains two trees. ~ One of these trees (called the tableau-tree) is built to create the restricted family R(N) of standard ~ tableaux, while the other tree (called the list-tree) 2 is the a list balanced of partitions [4] binary that tree have which been keeps created track by the of ibii algorithm. Ri The parameters A, A 2,..., A are read from the input file and two global arrays are constructed to 1 R 3 41 k 24 tableaux of record eachthe integer constraints thej, collection 1 jon Nthe R(N). which allowable This determine islocations accom- the Fig. 4. Tableau-tree in depth-first order. plished without actually generating the objects F( A, A2,..., At) and LF( X~,A2,..., At) themselves. The tableaux-tree is constructed by the proceand let R( N) be the collection of all T E ST( N) with the following properties: (i) Wheneverj + I is immediately to the right of ~ dure CONSTRUCT recursively using backtrack- ing. Since only the shape of a tableau that appears as a terminal node is required, the data structure in LF( A, A2,..., Ar), then j + 1 appears employed keeps track of the sizes of the parts of south east of j in ~. the tableaux being generated rather than the (ii) Whenever k is immediately below] in LF( A1, tableaux themselves. The allowable rows for A2,..., Ar), then k appears north west of j in T. branching at each particular node of the tableau- Then tree are passed as arguments in the next call to procedure CONSTRUCT. Once a terminal node {A )(A2)... (At) = ~ (P(r)). of the tableau-tree is reached, the rowlengths are T R( N) converted into a string to facilitate data manipula- As an example take A = (2 1), A2 = (12). Then tion at later stages of the program. The encoded LF( A, A2) is as shown in fig. 3. partition is then inserted into the list-tree which To expand (A ) (A2), we construct all standard maintains the list of partitions previously genertableaux in ST(5) such that ated together with the number of times each one was generated (multiplicity). If the partition al- 2 appears N W of 1, ready appears in the current list-tree then its mul- 3 appears S E of 2, tiplicity is incremented by one and control re- 5 appears N W of 4. turned node is to created the main and program. the tree Otherwise is balanced a if new necesleef This is done by constructing the tree, depicted in sary. fig. 4, in depth-first. order. Taking the underlying After the generation of the collection P( R( N)), shapes of the terminal nodes of this tree, we have the list-tree is traversed in depth-first order and the expansion the list of generated partitions together with their (2 l)(12) = (3 2) + (3 12) + (22 1) + (2 l~) multiplicities are written to a user-specified output file.

4 E~ecio~lu/ Outer products of Schur functions 3.2. Description of the procedures initializes the variables. The two global arrays that are used to determine R(N) are constructed as The names and brief descriptions of the procefollows: DOWN RIGHT [k]: =j if j is immediately dures used are as follows: above k in the lexicographic filling of the associated skew-shape, DECLARATIONS DOWN_ RIGHT [k]: = 0 if no such entry ex- In this procedure, the parameters that are used by ists; the main program SCHUR and their data types UP~LEFT [j + 1]: =j if j is immediately to are declared. The problem size is controlled by a the left of j + I in the lexicographic filling in single constant declared as MAXPROBSIZE. the associated skew-shape, SCHUR up_left [j + 1]: = 0 if no such entry exists. This is the main driver of the program. These assignments are made as the input partitions are read. OUTPUT Traverses the list-tree in depth-first order. The UPDATE_ TABLEAU multiplicity, the number of parts and the parts The current shape ROW_ LENGTHS is updated themselves of each partition are written to an and the available rows for the insertion of the next output file, element is recorded. The array HEIGHT marks the indices of the rows occupied by the numbers CONSTRUCT that are already in the tableau. In conjunction This procedure constructs the collection P( R( N)). with the information in the arrays DOWN - The input arguments are as follows: RIGHT and UP_ LEFT, the maximum and the (i) The element j to be added to the present minimum indices of the rows available for the next tableau. element are determined. (ii) The minimum and the maximum indices of the rows that j can be attached to as de- 33 1/0 format termined by the elements in LF(A, A 2 AT) which lie immediately above and to the left of The user is prompted to enter the names of the j (if any). Thus the present node in the input and output files. The default is the console tableau-tree can have at most (maximum device. The input file is expected to be a linear list minimum) subtrees. where the input partitions are formated as (iii) The parts of the underlying partition which can be incremented by one without violating k A, A 2... A,~(cr), the monotonicity condition of the rowlengths. where the entries are separated by blanks and <Cr) The indices of these rows are kept in the array denotes carriage return. GOOD_ ROWS. The partitions generated by the program are (iv) The array ROW- LENGTHS which stores the written to the specified output file in the form parts of the underlying partition. These variables must be initialized by the main m12 1 ~ I~~2 driver and passed as value parameters. Here m,~is the multiplicity of the Schur function INSERT Inserts a partition into the list-tree and balances the tree if necessary. SET_ CONSTRAINTS Reads the input from the specified input file and (p.) in the expansion of the product of the input Schur functions. If the output device is other than disk or the console then the write and writeln statements that appear in procedure OUTPUT should be modified accordingly.

5 0. E~ecio~lu/ Outer products of Schurfunctions Performance gramming language is ref. [6]. The UCSD Pascal system is described in detail in ref. [7]. The complexity of the algorithm is determined by the list-insertion and maintenance portion of the program which is of (S( n log n), where n is the Acknowledgements number of terminal nodes generated in the tableau-tree. The main body of the program which Special thanks go to Mark Erikson for coding actually generates the collection P(R(N)) can be and fine-tuning the present version of program altered to perform in linear time by judicious use Schur. The Electric Engineering and Computer of linked-list structures to pass the list of available Science department of UCSD kindly provided rows in the recursive calls to procedure CON- access to their computational facilities. STRUCT. Moreover, by suitably encoding the partitions created, bucket-sort can be applied to generate the output list in linear time. Therefore References theoretically the algorithm can be implemented in linear time. For large size problems though, the [II J. Remmel and R. Whitney, Multiplication of Schur Funcexponential growth of the partition function re- tions (to appear). [21D.E. Littlewood and A.R. Richardson, Phil. Trans. A 233 nders bucket-sort quite impractical in terms of (1934) 99. space requirements. [31D.E. Littlewood, The Theory of Group Characters, 2nd ed. (Oxford Univ. Press, London, 1950) p. 94. [41N. Wirth, Algorithms + Data Structures = Programs (Pren- 4. Remarks tice-hall, Englewood Cliffs, NJ, 1976). [51B.G. Wybourne, Symmetry Principles of Atomic Spectroscopy (John Wiley, New York, 1970). The correctness of this implementation has been [61 K. Jensen and N. Wirth, Pascal User Manual and Report checked extensively by making use of the tables of (Springer-Verlag, New York, 1975). outer products of Schur functions that appear in [71K.L. Bowles, Beginner s Manual for the UCSD Pascal ref. [5]. System (McGraw-Hill, Peterborough, New Hampshire, The standard reference of the Pascal pro- 1979).

MURNAGFL4N S RULE AND THE IRREDUCIBLE CHARACTERS OF THE SYMMETRIC GROUP. References

Computer Physics Communications 31(1984) 357 362 357 North-Holland, Amsterdam MURNAGFL4N S RULE AND THE IRREDUCIBLE CHARACTERS OF THE SYMMETRIC GROUP Omer EGECIOGLU and Geraldo M. COSTA Department of Mathematics,