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\def\irrep{irreducible representation}
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\noindent
{\sc S\'eminaire Lotharingien de Combinatoire}  {\rm B36h (5pp.)}
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\centerline{\bf Plethysm and the Non-Compact Groups Sp(2n,R)}
\bigskip
\centerline{B.G.Wybourne }
\medskip
\centerline{Instytut Fizyki, Uniwersytet Miko\l aja Kopernika}
\centerline{ul. Grudzi\c adzka 5/7}
\centerline{ 87-100 Toru\'n}
\centerline{Poland}
\bigskip
\noindent{\bf ABSTRACT} 
\item{} Some preliminary results on the plethysms for the non-compact group $Sp(2n,R)$ are presented. Complete results are given for power 2 plethysms of 
the two fundamental \irrep{s} of $Sp(2n,R)$. Several new $S-$function identities
arise from this work. The stabilisation properties of the plethysms are
briefly considered and some remarkable conjugacy mappings observed.
\bigskip





\noindent{\bf Introduction}
\smallskip

The plethysm of $S-$functions has been the subject of much research ever since
its introduction by Littlewood$^1$. Many
applications have been made to the classical compact Lie groups by expressing
the characters of the \irrep{s} of the group in terms of $S-$functions$^{2-6}$.
To date rather scant attention has been paid to application of plethysm to
non-compact Lie groups$^{7-9}$.

The non-compact group $Sp(2n,R)$ is of special interest in physics as it is the
dynamical group$^{10}$ of the $n-$dimensional isotropic harmonic oscillator which
finds important applications in symplectic models of nuclei$^{11}$ and in the
mesoscopic physics of quantum dots$^{12,13}$. The non-trivial unitary
\irrep{s} of $Sp(2n,R)$ are all of infinite dimension$^{14,15}$. An extensive
outline of notation, characters, Kronecker products and branching rules is
developed in reference 15. In other matters we follow the notation of Macdonald$^{16}$
Arbitrary positive discrete harmonic
series \irrep{s} of $Sp(2n,R)$ will be labelled as $<{k\over2};(\lambda)>$ or
equivalently as $<s\kappa;(\lambda)>$ where $\kappa$ and $s$ are the integer
and residue parts of ${k\over2}$.

The infinite set of states of a harmonic oscillator span the pair of infinite-
dimensional fundamental unitary \irrep{s} of $Sp(2n,R)$ which we shall
designate$^{15}$ as $<\o;(0)>$ and $<\o;(1)>$. Our central problem is to resolve
symmetrised powers of these two \irrep{s} which amounts to evaluating the
plethysms
$$s_\lambda(<s;(0)>)\qquad \hbox{and}\qquad s_\lambda(<s;(1)>)\eqno(1)$$

To proceed with the plethysm problem for $Sp(2n,R)$ we first consider the
$Sp(2n,R) \rightarrow U(n)$ decompositions where $U(n)$ is the unitary group in
$n-$dimensions and then show how these can be used to build up $Sp(2n,R)$
plethysms up to some finite cutoff and present some complete results for
$\lambda \vdash 2$. We are then led to some new $S-$function identities and
after some comments on the stability of $Sp(2n,R)$ plethysms we end with
a  remarkable observation of the existence of a mapping between the two types
of plethysms.

\bigskip

\noindent{\bf Sp(2n,R) $\rightarrow$ U(n) decompositions}
\smallskip

Under the restriction$^{15}$  $Sp(2n,R) \rightarrow U(n)$ a given \irrep\ of
\hfill\break $Sp(2n,R)$
decomposes into an infinite set of finite dimension \irrep{s} of the unitary
group $U(n)$. In the case of the two fundamental \irrep{s} of $Sp(2n,R)$ we
have$^{15}$
$$\eqalignno{<s;(0)>& \rightarrow \varepsilon^{\o} M_+&(2a)\cr
             <s;(1)>& \rightarrow \varepsilon^{\o} M_-&(2b)\cr}
$$
where $M_+$ and $M_-$ are the {\it even} and {\it odd} weight $S-$functions
$s_m$ appearing in the infinite series
$$M = \sum_{m = 0}^\infty s_m\eqno(3)$$
In general one has$^{15}$
$$<{k\over2};(\lambda)> \rightarrow \varepsilon^{{k\over2}}\cdot
((s_{\lambda_s})_N^k \cdot D_N)_N\eqno(4)$$
where $N = min(n,k)$ and $D$ is the infinite $S-$function series
$$D = \sum_\delta s_\delta\eqno(5)$$
where the $\delta$ are partitions involving only {\it even} parts. The
subscript $N$ means that all terms involving partitions into more than $N$ parts
are to be discarded. The first $\cdot$ indicates a product in $U(n)$ and
the second $\cdot$ a product in $U(N)$. $(s_{\lambda_s})^k$ is a {\it signed
sequence}$^{14,15}$ of terms $\pm s_\rho$ such that $\pm s_\rho$ is equivalent to
$s_\lambda$ under the modification rules of the orthogonal group $O(k)$.

\bigskip

\noindent{\bf Plethysms in Sp(2n,R)}
\smallskip

We are primarily interested in plethysms of the form  $s_\lambda(<s;(0)>)$
and $s_\lambda(<s;(1)>)$.
No general procedure seems to be known for evaluating $Sp(2n,R)$ plethysms.
Here we evaluate the terms, up to a given weight, by first decomposing the
$Sp(2n,R)$ \irrep\ into those of $U(n)$, performing the plethysm at the $U(n)$
level and then inverting to get \irrep{s} of $Sp(2n,R)$. This has been done
for all $\lambda \vdash 4$ and in some cases to $\lambda \vdash 6$. Tables of
the relevant plethysms are located at http://www.phys.uni.torun.pl/$\sim$bgw/.
In the case of $\lambda \vdash 2$ it is possible to obtain completely general
results as follows
$$\eqalignno{s_2(<s;(0)>) &= \sum_{i=0}^\infty <1;(0 + 4i)>\cr
             s_{1^2}(<s;(0)>)&= \sum_{i=0}^\infty <1;(2 + 4i)>\cr
             s_2(<s;(1)>)&= \sum_{i=0}^\infty <1;(2 + 4i)>\cr
             s_{1^2}(<s;(1)>)& = <1;(1^2)> + \sum_{i=0}^\infty <1;(4 + 4i)>&(6)\cr}
$$
These results imply that the following $S-$function identity must hold
$$s_{1^2}(M_+) = s_2(M_-)\eqno(7)$$
as indeed may be shown to be the case$^{17}$.

If $L_+$ and $L_-$ are respectively the positive and negative terms of the series
$$L = \sum_{m=0}^\infty (-1)^m s_m\eqno(7)$$
then one finds
$$s_{1^2}(L_+) = s_2(L_-)\eqno(8)$$
Still further identities arise for the infinite $S-$function series defined by
$$\eqalignno{A_\pm& = L_\pm(s_{1^2})\qquad B_\pm = M_\pm(s_{1^2})\cr
             C_\pm& = L_\pm(s_2)\qquad D_\pm = M_\pm(s_2)&(8)\cr}
$$
Use of the associativety property of plethysms leads directly to
$$s_{1^2}(Z_+) = s_2(Z_-)\eqno(9)$$
for $Z = A, B, C, D$. Furthermore
$$s_2(Z) = ZZ_+\qquad\hbox{and}\qquad s_{1^2} = ZZ_-\eqno(10)$$
The study of plethysms within the group $Sp(2n,R)$ leads to still further
identities. The observation that
$$s_{21^2}(<s;(0)>) = s_{31}(<s;(1)>)\eqno(11)$$
leads to the remarkable $S-$function identity
$$s_{21^2}(M_+) = s_{31}(M_-)\eqno(12)$$
which generalises to
$$s_\sigma(s_{1^2}(M_+)) = s_\sigma(s_2(M_-))\eqno(13)$$
Again these identities extend to the series $Z$ defined earlier.

\bigskip

\noindent{\bf Stability of Kronecker products and plethysms}
\smallskip

A given plethysm, Kronecker product or decomposition will be said to be
{\it stable} if at the stable value of $n = n_s$ there is a one-to-one
mapping between the resultant list of \irrep{s} obtained at the stable value
$n_s$ and those obtained for all values of $n > n_s$. The $Sp(2n,R)$
Kronecker product$^{15}$
$$<{k\over2}(\lambda)>\times<{\ell\over2}(\nu)> =
<{{(k + \ell)}\over2};((s_{\lambda_s})^k\cdot (s_{\nu_s})^\ell\cdot D)_{k + \ell,n}>
\eqno(14)$$
is certainly stable for all $n \geq (k + \ell)$. We say {\it certainly}
because in some cases {\it premature stability} may occur for values of
$n < (k + \ell)$.

One observes that the power 3 plethysms for the two fundamental \irrep{s}
stabilise at $n = 3$ which is consistent with the stabilsation of the
products $<s;(0)>\times<1;(\mu)>$ and $<s;(1)>\times<1;(\mu)>$ at $n = 3$
and for similar reasons stabilsation of power $N$ plethysms must occur at
$n = N$ as observed. Again, premature stabilisation for individual
plethysms may occur for $n < N$. Thus foe $N = 3$ all the plethysms stabilise
at $n = 2$ except for $s_{1^3}(<s;(1)>)$ which stabilises at $n = 3$.
Stabilisation for arbitrary $N$ occurs at $n = N - 1$ except for
$s_{1^N}(<s;(1)>)$ which stabilises at $n = N$.

\bigskip

\noindent{\bf Plethysms and conjugacy mappings}
\smallskip

Below we give two short examples of plethysms with terms kept to weight 10.

\setbox1 = \vbox{\settabs5\columns{
\+$s_4(<s;(0)>) =$&$<2;(0)>$&$+\ <2;(4)>$&$\ + <2;(4^2)>$&$\ + <2;(6)>$\cr
\+&$\ + <2;(62)>$&$\ + <2;(73)>$&$\ + 2<2;(8)>$&$\ + <2;(91)>$\cr
\+&$\ + <2;(10\ >$\cr}}
$$\box1$$
\setbox2 = \vbox{\settabs5\columns{
\+$s_{1^4}(<s;(1)>) =$&$<2;(1^4)>$&$\ + <2;(41^2)>$&$\ + <2;(4^2)>$&$\ + <2;(61^2)>$\cr
\+&$\ + <2;(62)>$&$\ + <2;(73)>$&$\ + 2<2;(81^2)>$&$\ + <2;(91)>$\cr}}
$$\box2$$
Looking at the above results one cannot help but be struck by the apparent simple
mapping between them. Indeed looking at much more extensive tabulations one
observes that the terms in $s_\lambda(<s;(0)>)$ are simply related to those
of $s_{\tilde\lambda}(<s;(1)>)$ by a one-to-one mapping subject to the
following simple rules:-
\setbox3= \vbox{\settabs6\columns{
\+$\lambda \vdash 2$&$(0) \rightarrow (1^2)$\cr
\+$\lambda \vdash 3$&$(0) \rightarrow (1^3)$&$(a) \rightarrow (a1)$&$(a1
) \rightarrow (a)$\cr
\+$\lambda \vdash 4$&$(0) \rightarrow (1^4)$&$(a) \rightarrow (a1^2)$&
$(a1^2) \rightarrow (a)$\cr
\+$\lambda \vdash 5$&$(0) \rightarrow (1^5)$&$(a) \rightarrow (a1^3)$&
$(ab) \rightarrow (ab1)$&$(ab1) \rightarrow (ab)$\cr
\+$\lambda \vdash 6$&$(0) \rightarrow (1^6)$&$(a) \rightarrow (a1^4)$&
$(a1^4) \rightarrow (a)$&$(ab) \rightarrow (ab1^2)$&$(ab1^2) \rightarrow (ab)$\cr
}}
$$\box3$$
The explanation of such simple results remains unknown and deserves further
study.

\bigskip

\noindent{\bf Concluding remarks}
\smallskip

The study of plethysms for the non-compact group $Sp(2n,R)$ throws up many
surprises that could be of interest to combinatorialists. The study of plethysms
for other non-compact groups, such as $SO(4,2)$ which plays a key role in
Coulomb systems, is completely unknown. I hope in these notes I might
stimulate others to consider some of the problems raised herein.

\noindent{Acknowlegements}
\smallskip

This work has been supported by a grant from the Polish KBN. All calculations
were made using SCHUR$^{18}$. I thank the seminar organisers for the opportunity
to participate in a stimulating and well organised meeting.


\bigskip

\noindent{\bf References}
\smallskip

\item{1.} D. E. Littlewood, {\it The Theory of Group Characters and Matrix
Representations of Groups} 2nd edn (Oxford: Clarendon) (1950)
\item{2.} D. E. Littlewood, Invariant theory, tensors and group characters
{\it Phil. Trans. R. Soc.} A{\bf239} 305-65 (1944)
\item{3.} D. E. Littlewood, On invariant theory under restricted groups
{\it Phil. Trans. R. Soc.} A{\bf239} 387-417 (1944)
\item{4.} B. G. Wybourne, {\it Symmetry Principles and Atomic Spectroscopy}
(New York: Wiley Interscience) (1970)
\item{5.} R. C. King, Luan Dehuai and B. G. Wybourne, Symmetrised powers of
rotation group representations {\it J. Phys. A: Math. Gen. \bf14} 2509-38 (1981)
\item{6.} S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins, Normal
forms for tensor polynomials: I. The Riemann tensor
{\it Class. Quantum Grav. \bf9} 1151-97 (1992)
\item{7.} M. Yang and B. G. Wybourne, New $S-$function series and non-compact
Lie groups {\it J. Phys. A: Math. Gen. \bf19} 3513-25 (1986)
\item{8.} M. J. Carvalho, Symmetrised Kronecker products of the fundamental
representation of $Sp(n,R)$ {\it J. Phys. A: Math. Gen. \bf23} 1909-27 (1990)
\item{9.} B. G. Wybourne, The representation space of the nuclear
symplectic $Sp(6,R)$ shell model {\it J. Phys. A:Math. Gen. \bf 25} 4389-98
(1992)
\item{10.} B. G. Wybourne, {\it Classical Groups for Physicists} (New York:Wiley)
(1974)
\item{11.} D. J. Rowe, Microscopic theory of the nuclear collective model
  {\it Rep. Prog. Phys. \bf48} 1419-80   (1985)
\item{12.} R. W. Haase and N. F. Johnson, Schur function analysis of the unitary
discrete series representations of the non-compact symplectic  group
 {\it J. Phys. A:Math. Gen. \bf 26} 1663-72 (1993)
\item{13.} B. G. Wybourne, Applications of $S-$functions to the quantum Hall
effect and quantum dots {\it Rept. Math. Phys. \bf34} 9-16 (1994)
\item{14.} D. J. Rowe, B. G. Wybourne and P. H. Butler, Unitary representations,
branching rules and matrix elements for the non-compact symplectic groups
{\it J. Phys. A:Math. Gen. \bf 18} 939-53 (1985)
\item{15.} R. C. King and B. G. Wybourne, Holomorphic discrete series and
harmonic series unitary \irrep{s} of non-compact Lie groups : $Sp(2n,R),
U(p,q)$ and $SO^\ast(2n)$
{\it J. Phys. A:Math. Gen. \bf 18} 3113-39 (1985)
\item{16.} I. G. Macdonald, {\it Symmetric functions and Hall Polynomials}
(Oxford: Clarendon) 2nd edn (1995)
\item{17.} K. Grudzinski and B. G. Wybourne, Plethysm for the non-compact group
$Sp(2n,R)$ and new $S-$function identities,
{\it J. Phys. A:Math. Gen.} (submitted)
\item{18.} See http://smc.vnet.net/Christensen.html or http://www.phys.uni.torun.pl/$\sim$bgw/

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