Jean-Yves Thibony, Frederic Toumazety and Brian G Wybournez x. Abstract. We give the decomposition of the Kronecker products and the

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1 Products and plethysms for the fundamental harmonic series representations of U p; q) Jean-Yves Thibony, Frederic Toumazety and Brian G Wybournez x yinstitut Gaspard Monge, Universite de Marne-la-Vallee, 2 rue de la Butte-Verte, Noisy-le-Grand cedex, France zinstytut Fizyki, Uniwersytet Miko laja Kopernika, ul Grudziadzka 5/7, 87-1 Torun, Poland Abstract. We give the decomposition of the Kronecker products and the symmetrized Kronecker squares of all the fundamental representations of the harmonic series of unitary irreducible representations of Up; q). The results for U2; 2) are relevant to two-electron hydrogenic like atoms. 1. Introduction Bohr, in his very rst paper [3], on what has become known as \The Bohr-model" of the atom, made the surprising discovery that the energies of levels of the non-relativistic hydrogen atom could be expressed in appropriate units) as simply E n = 1 n 2 with n = 1; 2; : : : With the advent of the Schrodinger equation for the H-atom it became apparent that each value of n could be associated with orbital angular momenta of ` = 1; 2; : : : ; n 1 and associated with each value of ` there were 2` + 1) values of the angular momentum projection eigenvalues m` leading to each energy level E n being associated with n 1) 2 eigenfunctions. Initially such a high degeneracy appeared surprising. Pauli [7] noted that in a purely Coulombic central eld there was an additional constant of the motion associated with the Runge-Lenz vector and from there was led to the realisation that the observed degeneracies were precisely the dimensions of certain of the irreducible x Supported by a Polish KBN Grant

2 2 representations of the group SO4) SU2) SU2), in particular those commonly designated as [n 1; ] fn 1g fn 1g. Much later, Barut and Kleinert [2] observed that all the discrete levels of a H- atom spanned a single innite dimensional irreducible representation of the non-compact group SO4; 2) SU2; 2) with the group being referred to as the dynamical group of the H-atom [2, 9]. The Runge-Lenz vector ceases to be a constant of the motion for two or more electrons in a central Coulomb eld [2, 9, 1, 4] and the SO4) symmetry is broken. Nevertheless, it can be useful to consider the n electron states starting with the single irreducible representation of SU2; 2), or more simply U2; 2), and then forming symmetrized n fold tensor products which will be the central problem considered here. For greater generality we shall initially consider the group Up; q) as previously studied [6] by King and Wybourne. After a brief sketch of the relevant properties of Up; q) we tackle the problem of resolving the Kronecker powers of the relevant irreducible representation into its relevant symmetrized powers, namely the problem of plethysms in Up; q). In the process we are able to give closed results for the second powers of the fundamental harmonic series irreducible representations of Up; q) which thus yields, in the case of two electrons, the appropriate spin triplet and singlet states. 2. The fundamental harmonic series irreducible representations of Up; q) Following [6], we may embed the non-compact group Up; q) in Sp2p + 2q; R) whose harmonic representation ~ decomposes as where ~! H = H + 1 m=1 H m + H m ) 1) H = f1; )g 2a) H m = f1; m)g m = 1; 2; : : : 2b) H m = f1 m; )g m = 1; 2; : : : 2c) Upon restriction to the maximal compact subgroup Uq) Up) we have H = f1; g! ") H m = f1; m)g! ") H m = f1 m; )g! ") f kg fkg) f kg fm + kg) fm + kg fkg) 3a) 3b) 3c)

3 The harmonic series unirreps fk; )g of Up; q) are generated by considering powers [5] H k of H. Under restriction from Upk; qk) to Up; q) Uk) H! ; fk; )g f; g 4) where the partition ) has not more than p parts and ) not more than q parts and their conjugate partitions ~) and ~) satisfy the constraints [5] ~ 1 + ~ 1 k 5a) ~ 1 p and ~ 1 q 5b) 3 3. Kronecker products for all of the fundamental harmonic series unirreps The Kronecker product of two arbitrary unirreps of Up; q) may be evaluated following [6] to give fk; )gf`; )g = fk +`f s g k f s g` f g; f s g k f s g` fg))g6) where the notation is as in [6] and it is understood that ; ) if 1 ~ p; ~ 1 q and ; )) k+`;p;q = ~ 1 + ~ 1 k + ` otherwise Specialization of 6) to the fundamental harmonic series of Up; q) yields the following cases 7) H 2 = H 2 m = H 2 m = H m H m = H r H s = H r H s = 1 n= 1 n= 1 n= 1 minr;s) x= minr;s) x= f2n; n)g H r H s = f2r; s)g + f2n; n + 2m)g + f2n + 2m; n)g + f2m + k; m + k)g m p=1 m p=1 f2; r + s x; x)g + f2r + s x; x; )g + 1 8a) f2; 2m p; p)g m > 8b) f22m p; p; )g 1 1 8c) 8d) f2 k; r + s + k)g 8e) f2r + s + k; k)g 8f) f2r + k; s + k)g r; s > 8g)

4 4 H H m = f2; m)g + H H m = f2m; )g f2 k; m + k)g 8h) f2m + k; k)g 8i) 4. Symmetrized squares of the fundamental harmonic representations To separate the Kronecker squares of the representations H m of Up; q) into its symmetric and antisymmetric parts, we rst solve the corresponding problem for the complete harmonic representation H. This is done by restricting the H of U2p; 2q) through the chain U2p; 2q) Up; q) U2) Up; q) S 2 Up; q) : 9) Under U2p; 2q) # Up; q) U2), we know that H! ~ 1 + ~ 1 2 f2; )g f; g : 1) Therefore, we just have to determine the restriction to S 2 of the U2) representations f; g. It is known [1], see also [8]) that the Frobenius characteristic of the decomposition of fmg under Uk) # S k is the coecient of z m in the series h k 1 z = ky j=1 1 1 z j `k ~K ;1 kz)s 11) where K;1 ~ kz) are the cocharge) Kostka-Foulkes polynomials. In particular for k = 2, fmg # S 2 is the coecient of z m in 1 [2) + z11)] 12) 1 z)1 z 2 ) so that fmg! p 2 m)2) + p 2 m 1)11) 13) where p 2 m) is the number of partitions of m into parts not greater that 2, that is, p 2 m) = d m+1e. 2 Taking into account the U2) equivalences f; 1 2 g 2 f 1 2 g, f m; ng m fn + mg and f 1 2 ; g 1 f 1 2 g, we obtain p2 1 2 )2) + p )11) for 2 even f; 1 2 g! p )2) + p )11) for 2 odd 14a) p2 m + n)2) + p 2 m + n 1)11) for m even f m; ng! 14b) p 2 m + n 1)2) + p 2 m + n)11) for m odd p2 1 2 )2) + p )11) for 1 even f 1 2 ; g! 14c) p )2) + p )11) for 1 odd

5 5 Now, we have H f2g 1 H + m + H m ) m1h = H f2g + H m + H m ) 1 H m m1h m1 + H r H s 1 H m m1 r;s1 = H f2g + m1 H m H m + R : To extract H f2g from H f2g, we remark that since under Up; q) # Up) Uq) H! ) mf mg fmg the Kronecker square of H can only contain terms whose restriction to Up) Uq) is a sum of representations 2 )f; g such that jj = jj. Clearly, the terms in R are not of this form, and to obtain H f2g, we just need to compute the terms of the form f2 m; m)g in H f2g and to remove the contribution of P m1 H m H m. We know from the above discussion that the multiplicity of f2 m; m)g in H f2g is equal to p 2 m + m) = m + 1 for m even, and to p 2 m + m 1) = m for m odd. On the other hand, H m H m = kf2m + k; m + k)g so that a given f2 m; m)g occurs exactly m times in P k1 H k H k. Removing this contribution, we are left with H f2g = kf22k; 2k)g : 15) Since H 2 = P mf2 m; m)g, we also have H f1 2 g = kf22k + 1; 2k + 1)g : 16) To split the square of H m m 1), we rst observe that under restriction to Up) Uq), it yields a sum of representations of the form 2 )fg fg such that jj = jj + 2m. Next, we proceed as above to extract it from H f2g. We have H f2g Hm + j6=m H j 1 = H m f2g + H m H j + 1 H m j

6 6 1 H m j 1 H m+k A 1 H m+j k1 Therefore, to extract H m f2g, we just have to select from H f2g the terms having the correct restriction property to Up) Uq) and to subtract the contribution of the crossed products H m j H m+j j 1). Suppose rst that m 1. Then, H m j H m+j = H H 2m + The terms of this sum are m 1 r=1 H r H 2m r + r1 H r H 2m+r : 17) H H 2m = kf2 k; 2m + k)g ; 18a) r H r H 2m r = 2m i; i)g + f2 k; i=1f2; 2m + k)g ; 18b) k H r H 2m+r = kf2r + k; 2m + r + k)g 18c) so that H m j H m+j = m 1 i=1 m i)f2; 2m i; i)g + km + k)f2 k; 2m + k)g : 19) Now, the multiplicity of f2; 2m i; i)g in H f2g is p 2 2m 2i) = m i+1 for i even, and p 2 2m 2i 1) = m i for i odd. Similarly, the multiplicity of f2 k; 2m + k)g in H f2g is equal to p 2 2m + 2k) = m + k + 1 for k even, and to p 2 2m + 2k 1) = m + k for k odd. Finally, we are left with H m f2g = bm=2c i=1 Similarly, we obtain H m f1 2 g = Likewise, H m f2g = H m f1 2 g = f2; 2m 2i; 2i)g + kf22k; 2m + 2k)g : 2a) bm 1)=2c i= bm=2c i=1 f2; 2m 2i 1; 2i + 1)g + kf22k + 1; 2m + 2k + 1)g 2b) f22m 2i; 2i; )g + kf22m + 2k; 2k)g ; 2c) bm 1)=2c i= f22m 2i 1; 2i + 1; )g + kf22m + 2k + 1; 2k + 1)g 2d)

7 7 5. Conclusion We have been able to obtain complete results for all the Kronecker products, and their symmetrized squares, for all the fundamental harmonic unirreps of Up; q) expressing them in a compact closed form. The plethysms of the square of the unirrep H for U2; 2) give the complete set of U2; 2) unirreps that arise in a two-electron hydrogenic-like atom with the symmetric part describing the spin singlets S = ) and the antisymmetric part the spin triplets S = 1). The groundstate 1s 2 1 S) is the rst level of an innite tower of states associated with the f2; )g unirrep while the lowest 3 SP level is the rst level of an innite tower associated with the f21; 1)g unirrep. A complete account of the twoelectron hydrogen like states remains to be considered but knowing the relevant U2; 2) unirepps is a signicant beginning. For an n electron hydrogen-like atom n > 2) the resolution of plethysms of the type H fg ` n) is a formidable task and complete results of the type considered herein cannot be expected. References [1] Aitken A C 1946 Proc. Edinburgh Math. Soc. 2) [2] Barut A O and Kleinert H 1967 Phys. Rev [3] Bohr N 1913 Phil. Mag. 476 [4] Butler P H and Wybourne B G 197 J. Math. Phys [5] Kashiwara M and Vergne M 1978 Inventiones Math [6] King R C and Wybourne B G 1985 J. Phys. A: Math. Gen [7] Pauli W 1926 Z. Phys [8] Scharf T, Thibon J-Y and Wybourne B G 1993 J. Phys. A: Math. Gen [9] Wybourne B G 1974 Classical groups for physicists New-York: Wiley) [1] Wulfman C E 1971 Group theory and its applications Vol II E M Loebl Ed New York: Academic Press) pp