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%I #8 Aug 30 2019 04:08:40
%S 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,
%T 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,
%U 4,12,18,12,6,12,16,8,10,10
%N Array used to find the eigenvalues of the quadratic Casimir operator for the Lie algebras A_n = su(n+1), n>=1.
%C 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.
%C 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.
%C The numbers in the even numbered rows are divisible by 2.
%C 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.
%C 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)).
%C The formula used to compute the C2_n polynomials can be found, e.g., in the van Ritbergen et al. reference, eq. (27).
%C The author thanks Stephan Rachel and Ronny Thomale for drawing his attention to this exercise.
%C 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.
%C The sequence of row lengths is n*(n+3)/2=A000096(n)=[2,5,9,14,20,27,35,44,54,65,...], n>=1.
%C The row sums are conjectured to be 3*A002415(n+1)= [3, 18, 60, 150, 315, 588, 1008, 1620, 2475, 3630,...]
%D R. Slansky, Group theory for unified model building, Physics Reports, 79, No 1 (1981) 1-128.
%D T. van Ritbergen, A. N. Schellekens, J. A. M. Vermaseren, Group Theory for Feynman Diagrams, Int. J. Mod. Phys. A, 14 (1999), 41-96.
%H W. Lang: <a href="/A116863/a116863.txt">First 10 rows.</a>
%F 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]].
%F a(n,m) are the coefficients of the polynomial C2_n = C2_n(a[1],...,a[n]) in the above mentioned order.
%e [1,2],[2,2,2,6,6],[3,4,2,4,4,3,12,16,12],...
%e Row n=3 stands for the polynomial C2_3=
%e 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].
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Mar 24 2006
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