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A116863
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Array used to find the eigenvalues of the quadratic Casimir operator for the Lie algebras A_n = su(n+1), n>=1.
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1
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1, 2, 2, 2, 2, 6, 6, 3, 4, 2, 4, 4, 3, 12, 16, 12, 4, 6, 4, 2, 6, 8, 4, 6, 6, 4, 20, 30, 30, 20, 5, 8, 6, 4, 2, 8, 12, 8, 4, 9, 12, 6, 8, 8, 5, 30, 48, 54, 48, 30, 6, 10, 8, 6, 4, 2, 10, 16, 12, 8, 4, 12, 18, 12, 6, 12, 16, 8, 10, 10
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OFFSET
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1,2
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COMMENTS
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Irreducible representations of the rank n Lie algebra A_n (also called su(n+1)) are determined by their Dynkin labels (indices) [a[1],..,a[n]] which are nonnegative numbers. The quadratic Casimir operator C2_n for A_n is given (up to normalization) by an integer polynomial of degree 2 in these n labels.
The first n*(n+1)/2= A000217(n) numbers in row n give the coefficients of the quadratic terms according to the index ordering: 11,12,13,...,1n;22,23,...,2n;33,...,3n;...;nn. The last n numbers in row n give the linear terms according to the ordering: 1,2,...,n.
The numbers in the even numbered rows are divisible by 2.
The determinant of the Cartan matrix for the Lie algebra A_n is n+1. The factor 1/(n+1) from the inverse Cartan matrix has been taken out in the formula given below.
For the n X n Cartan matrix C(n) for the Lie algebra A_n see, e.g., the Slansky reference p. 81, table 6 (called A(A_n) there) and p. 82, table7, where the inverse Cartan matrix is given (called there G(A_n)).
The formula used to compute the C2_n polynomials can be found, e.g., in the van Ritbergen et al. reference, eq. (27).
The author thanks Stephan Rachel and Ronny Thomale for drawing his attention to this exercise.
The C2_n polynomial for A_n has n*(n+3)/2=A000096(n) terms: n*(n+1)/2= A000217(n) quadratic terms and n linear terms.
The sequence of row lengths is n*(n+3)/2=A000096(n)=[2,5,9,14,20,27,35,44,54,65,...], n>=1.
The row sums are conjectured to be 3*A002415(n+1)= [3, 18, 60, 150, 315, 588, 1008, 1620, 2475, 3630,...]
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REFERENCES
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R. Slansky, Group theory for unified model building, Physics Reports, 79, No 1 (1981) 1-128.
T. van Ritbergen, A. N. Schellekens, J. A. M. Vermaseren, Group Theory for Feynman Diagrams, Int. J. Mod. Phys. A, 14 (1999), 41-96.
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LINKS
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FORMULA
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The polynomials are C2_n:= (rowvec(n,2+a)*C^{(-1)}(n)*colvec(n,a))*(n+1), with C^{(-1)}(n) the inverse of the n X n Cartan matrix C(n) for A_n. Here colvec(n,a) is the transposed of [a[1],a[2],...,a[n]] and rowvec(n,2+a):=[2+a[1],2+a[2],...,2+a[n]].
a(n,m) are the coefficients of the polynomial C2_n = C2_n(a[1],...,a[n]) in the above mentioned order.
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EXAMPLE
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[1,2],[2,2,2,6,6],[3,4,2,4,4,3,12,16,12],...
Row n=3 stands for the polynomial C2_3=
C2_3(a[1],a[2],a[3])=3*a[1]^2+4*a[1]*a[2]+2*a[1]*a[3]+4*a[2]^2+4*a[2]*a[3]+3*a[3]^2+12*a[1]+16*a[2]+12*a[3].
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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