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\pageno=63
\auteurcourant={S.C. MILNE}
\titrecourant={UMBRAL CALCULUS}

\eightpoint
\leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1984$, 229/S--08}
\leftline{Actes 8\emini\ S\'emininaire Lotharingien, p. 63--70}

\tenpoint
\let\it=\sl

\vglue 1.5cm
\centerline{{\bf AN UMBRAL CALCULUS FOR POLYNOMIALS}}
\vskip 5pt
\centerline{{\bf CHARACTERIZING $U(n)$ TENSOR PRODUCTS}}
\vskip 2.8mm
\centerline{\sevenrm BY}
\vskip 2.8mm
\centerline{{\petcap Stephen} C. MILNE \footnote{$(^*)$}{Partially
supported by a Sloan Foundation Fellowship and
NSF grant MCS--8102032}}
\vskip 3mm
This talk reports on work to appear in [1] that has been done
jointly with L.C. \pd BIEDENHARN  and R.A. \pc GUSTAFSON|.

\def\gb{{_\mu G_q^{(n)}}}
\def\gbm{{_\mu ^m G_q^{(n)}}}

In this lecture we continue the study of the connection
between the invariant polynomials
  $$\eqalign{\gb&\equiv \gb\bigl((x_{ij}+\Delta _1)\bigr)\cr
&\equiv \gb(\Delta _1,\ldots,\Delta _n;x_{12},x_{23},
\ldots,x_{n-1,n},x_{n,1}).\cr}\leqno(1)$$
characterizing $U(n)$ tensor operators 
$\langle p,q,\ldots,q,0,\ldots,0\rangle $, and the classical theory of symmetric functions as presented in [9], that
was established in [5]. The polynomial $\gb(X)$ arise naturally in the
application of symmetry groups to mathematical physics. One such problem, with applications to spectroscopy at all levels, is the
construction of a suitable basis for the set of all bounded operators mapping the set of all unitary irreducible
representation spaces of the group into itself. The precise
problems that give rise to $\gb(X)$ are motivated in more detail
and put into~a broader mathematical setting in [2--4].

The above irrep label $\langle p,q,\ldots,q,0,\ldots,0\rangle $
consists of one $p$, $\mu $ $q$'s, and $n-\mu -1$ 0's.
Furthermore, $q$ determines $p$ since $\Delta _1+\cdots+\Delta
_n=p+\mu q$ and we are given $q$, $\Delta _1+\cdots+\Delta
_n$ and $\mu $.

Recently in [5--7] an alternate method for explicitly writing
down $\gb(X)$ in polynomial form has been given. These methods
provide a direct connection between the classical theory of
symmetric functions as presented in [9], and the symmetries
satisfied by a family of $U(n)$ invariant polynomials even more
general then $\gb(X)$. In [5] it is shown that after the change
of variables
  $$\Delta _i=\gamma _i-\delta _i\qquad x_{i,i+1}
=\delta _i-\delta _{i+1},\leqno(2)$$
$\gb(,\Delta _i,;,x_{i,i+1},)$ becomes an integral linear
combination of products of Schur functions 
$S_\alpha (,\gamma _i,)\cdot S_\beta (,\delta _i,)$ 
in the variables $\{\gamma _1,\ldots,\gamma _n\}$ 
and $\{\delta _1,\ldots,\delta _n\}$,
respectively. (See equation~(6) for a definition of
$S_\lambda (x_1,\ldots,x_n)$.) That is, it is directly proved that
$\gb(,\Delta _i,;,x_{i,i+1},)$ is a bisymmetric polynomial in the
variables $\{\gamma _1,\ldots,\gamma _n\}$ and $\{\delta
_1,\ldots,\delta _n\}$ with integer coefficients. This motivated
the study in [5] of the yet more  general bisymmetric polynomials
  $$\gbm(\gamma ;\delta )\equiv 
\gbm(\gamma _1,\ldots,\gamma _n;\delta _1,\ldots,\delta
_m)\leqno(3)$$ 
which are a common generalization of (2.2b) of [2] and equation
(2.17) of [10] with no numerator parameters. These polynomials
are given by

\rem Definition 4|Given that ${_\mu ^mG_0^{(n)}}(\gamma ;\delta
)\equiv 1$, we uniquely determine $\gbm(\gamma ;\delta )$ by
means of
  $$\displaylines{(5)\quad\gbm(,\gamma _i,;,\delta _i,)
=\sum _{\scriptstyle S\subset  I_n \atop 
\scriptstyle \|S\|=\mu +1}
(-1)^{\mu +1+\Sigma (S)}
\prod _{\scriptstyle i<j \atop \scriptstyle i\in S,\,j\in S^c}
(\gamma _i-\gamma _j)^{-1}\hfill\cr
\hfill{}\times \prod _{\scriptstyle i<j 
\atop \scriptstyle i\in S^c,\,j\in S}
(\gamma _i-\gamma _j)^{-1}
\prod _{\scriptstyle i=1 \atop \scriptstyle i\in S}^n
\prod _{l=1}^m (\gamma _i-\delta _l)
\cdot{_\mu ^mG_{q-1}^{(n)}}
(,\gamma _i-\chi (i\in S),;,\delta _i,),\quad\cr}$$
where $S\subset  I_n$ is a $(\mu +1)$-element subset of
$\{1,2,\ldots,n\}$, $\Sigma (S)$ denotes the sum of the
elements in $S$, and $\chi (A)$ is 1 if statement $A$ is true
and 0, otherwise.

\vskip 5pt
At this point we need to review some basic facts
about the Schur functions $S_\lambda $. Let
$\lambda =(\lambda _1,\lambda _2,\ldots,\lambda _r,\ldots\,)$ be
a {\it partition}, i.e., a (finite or infinite) sequence
of nonnegative integers in decreasing order,
$\lambda _1\ge \lambda _2\ge \cdots\ge \lambda _r\cdots$ such
that only finitely many of the $\lambda _i$ are nonzero. The
number of nonzero $\lambda _i$, denoted by $l(\lambda )$, is
called the {\it length} of $\lambda $. If $\sum \lambda _i=n$,
then $\lambda $ is called a {\it partition of weight} $n$,
denoted by $\left|\lambda \right|=n$. Given a partition $\lambda
=(\lambda _1,\ldots,\lambda _n)$ of length ${}\le n$, the Schur
fonctions $S_\lambda $ are defined by
  $$S_\lambda (x_1,\ldots,x_n)
={\det\bigl(x_i^{\lambda _j+n-j}\bigr)_{1\le i,j\le n}
\over \det\bigl(x_i^{n-j}\bigr)_{1\le i,j\le n}}.\leqno(6)$$
The determinant in the numerator of (6) is divisible in
${\bf Z}[x_1,\ldots,x_n]$ by each of the differences $(x_i-x_j)$,
$1\le i<j\le n$, and hence by their product, which is the
{\it Vandermonde determinant}
  $$\leqalignno{\prod _{1\le i<j\le n}(x_i-x_j)
&=\det\bigl(x_i^{n-j}\bigr)_{1\le i,j\le n}&(7{\rm a})\cr
&\equiv V_n(x_1,\ldots,x_n).&(7{\rm b})\cr}$$
Thus, the quotient in (6) is a symmetric polynomial in
$x_1$, \dots , $x_n$ with coefficient in {\bf Z}. For
example, $S_{(n)}=h_n$ and $S_{(1^n)}=e_n$ where
$h_n$ and $e_n$ are, respectively, the homogeneous and
elementary symmetric functions of $x_1$, \dots , $x_n$.

If $Z=m_1+\cdots +m_n$, with $m_i$ monomials, then
  $$S_\lambda (Z)\equiv S_\lambda (m_1+\cdots+m_n)
\equiv S_\lambda (x_1,\ldots,x_n)
\Big|_{\textstyle x_i=m_i}.\leqno(8)$$
On the other hand, $S_\lambda \bigl(-(Z)\bigr)$ 
is defined symbolically by
  $$S_\lambda \bigl(-(Z)\bigr)\equiv 
S_\lambda \bigl(-(m_1+\cdots+m_n)\bigr)
\equiv (-1)^{|\lambda |}S_{\lambda '}(Z),\leqno(9)$$
where $\left|\mu \right|=\lambda _1+\cdots+\lambda _n$ is the 
sum of the parts of $\lambda $ and $\lambda '$ is the conjugate
partition to $\lambda $. That is, $\lambda '=(\lambda '_1,\lambda
'_2,\ldots,\lambda '_{(\lambda _1)})$, with 
$\lambda '_i=\|\{j\mid \lambda _j\ge i\}\|$. For example $(5,2,1)$
is the conjugate partition of $(3,2,1,1,1)$. Note that 
$S_\lambda \bigl(-(m_1+\cdots+m_n)\bigr)$ is {\it not} equal to
$S_\lambda (x_1,\ldots,x_n)|_{x_i=-m_i}$. The definitions given
by (8) and (9) are implicit in [9 ; see Remark (3.10) of p.~26].

If $\rho =(\rho _1,\ldots,\rho _k)$, where $\rho _i$ are integers, then denote
by $(X)^\rho $
the monomial $x_1^{\rho _1}\ldots x_k^{\rho _k}$
where $X=\{x_1,\ldots,x_k\}$. Let the permutations
$w\in S_k$ act on subscripts. For example,
$w(X)^\rho =x_{w(1)}^{\rho _1}\ldots x_{w(k)}^{\rho _k}$.
Finally, let $\delta _k$ be the partition
$(k-1,k-2,\ldots,0)$ and $m^l$
the partition consisting
of $l$ parts equal to~$m$.

Fix $\mu $ and assume that $m-n\equiv \nu $ is a constant. 
Denote the
sets of variables $\{\gamma _1,\ldots,\gamma _{\mu +1}\}$,
$\{\gamma _{\mu +2},\ldots,\gamma _n\}$,
$\{\gamma _1,\ldots,\gamma _n\}$,
and $\{\delta _1,\ldots,\delta _m\}$
by $A$, $B$, $E$ and $F$, respectively. We then have
from [5] the following fundamental

\th Theorem 10|Let $\gbm(E;F)$ be defined as in $(5)$. We then 
have
  $$\displaylines{(11)\quad\gbm(E;F)
={(-1)^{\mu +1\choose 2} \over  V_n(E)}
\sum _{\scriptstyle \lambda 
=(\lambda _1,\ldots,\lambda _{\mu +1}) \atop \scriptstyle
\lambda _1\le m} S_\lambda (-F)\hfill\cr
\hfill{}\cdot\Biggl\{\sum _{w\in S_n}\epsilon (w)\cdot w
\Biggl[(A)^{m^{\mu +1}
+\delta _{\mu +1}-(\lambda _{\mu +1},\ldots,\lambda _1)}
(B)^{\delta _{n-(\mu +1)}}\quad\cr
\hfill{}\cdot
{_\mu ^mG_{q-1}^{(n)}}
(\gamma _1-1,\ldots,\gamma
_{\mu +1}-1,B;F)\Biggr]\Biggr\},\cr}$$  
where, without loss of
generalitly, $w$ acts only on $\gamma _1$, \dots, $\gamma _{\mu
+1}$ when applied to ${_\mu ^mG_{q-1}^{(n)}}$, and $\epsilon
(w)$ is the sign of the permutation $w$.
\finth

Starting with (11) and making direct use of the new
symmetries discovered in [5], it is shown in [1] that the 
bisymmetric polynomials $\gbm(E;F)$ are an integral linear
commbination of Schur functions $S_\lambda (E-F)$ in the symbol $E-F$, where $E-F$
denotes the difference of the two sets of variables $E$ and $F$.
Making use of properties of skew Schur functions 
$S_{\lambda /\mu }$ and $S_\lambda (E-F)$ one puts together an
umbral calculus for $\gbm(E;F)$. That is, working entirely with
polynomials, one uniquely determines $\gbm$ from 
${_\mu ^mG_{q-1}^{(n)}}$ and combinatorial rules (such as the
Littlewood-Richardson rule~[9]) involving Ferrers diagrams (i.e.
partitions). The deepest part of this umbral calculus is a
summation theorem, involving many different aspects of the
theory of Schur functions, which essentially reduces the double
sum in (11) to the single term in which $\lambda $ is the empty
partition and $w$ is the identity permutation. Here, we recall a
general theorem form [1] which illustrates how the structure of
$\gbm(E;F)$ ``stabilizes'' as $\nu $ increases while $m-n=\nu $
remains fixed, state the umbral claculus for $\gbm(E;F)$, and
just give the final formulas for ${_\mu ^mG_1^{(n)}}(E;F)$ and
${_\mu ^mG_2^{(n)}}(E;F)$ which are a direct consequence of the
umbral calculus and work in [5].

Before giving these formulas we need some more notation.
Let $\lambda $, $\rho $ be partitions. The skew Schur function
$S_{\lambda /\rho }$ is defined by
  $$S_{\lambda /\rho }(E)
=\sum _{\nu \subset  \lambda } c_{\rho
\nu }^\lambda \cdot S_\nu (E),\leqno(12)$$ 
where $\nu \subset  \lambda $ means $\nu _i\le \lambda _i$ for
all $i\ge 1$, and the integers $c_{\rho \nu }^\lambda $ are the
Littlewood-Richardson rule coefficients determined by
  $$S_\rho (E)\cdot S_\nu (E)
=\sum _\nu  c_{\rho \nu }^\lambda  S_\lambda (E).\leqno(13)$$
Note that $S_{\lambda /0}=S_\lambda $, 
where 0 denotes the zero partition. Also,
$c_{\rho \nu }^\lambda =0$ unless 
$\left|\lambda \right|=\left|\rho \right|+\left|\nu \right|$,
so that $S_{\lambda /\rho }$ is homogeneous of degree
$\left|\lambda \right|-\left|\rho \right|$, and is zero if
$\left|\lambda \right|<\left|\rho \right|$.
In fact, $S_{\lambda /\rho }=0$ unless $\rho \subset  \lambda $.

Given the sets of variables $E$ and $F$, let
$E+F$ be the set union
$\{\gamma _1,\ldots,\gamma _n,\delta _1,\ldots,\delta _m\}$.
There is then the following classical result
  $$S_\lambda (E+F)=\sum _{\rho \subset  \lambda } 
S_{\lambda /\rho }(E) S_\rho (F).\leqno(14)$$
It is immediate from (9) and (14) that
  $$\leqalignno{S_\lambda (E-F)
&=\sum _{\rho \subset  \lambda } S_{\rho /\lambda }(E)S_\rho
(-F)&(15{\rm a})\cr
&=\sum _{\rho \subset  \lambda } (-1)^{|\rho |} 
S_{\rho /\lambda }(E)\cdot S_{\rho '}(F),&(15{\rm b})\cr}$$ 
where $\rho '$ is the conjugate partition to $\rho $. The above
classical formulas can be found in Chapter~I of~[9].

We now state a fundamental result of [1].

\th Theorem 16|Fix $\mu \ge 0$ and $k=m-n$. There are integers
$a_{\lambda ,q}$, $\lambda $ a partition, $q$ a non-negative 
integer such that
  $$\gbm(E;F)
=\sum _\lambda  a_{\lambda ,q}S_\lambda (E-F),\leqno(17)$$
where $\left|\lambda \right|\le (\mu +1)(\mu +1+k)q$ and
$\lambda \subset  [(\mu +1+k)q]^{(\mu +1)q}$.

If $n\ge (\mu +1)q$ and $m\ge (m+1+k)q$, then $a_{\lambda ,q}$
is independent of $n$. More generally, if 
$S_\lambda (E-F)\not= 0$,
then $a_{\lambda ,q}$ is independent of $n$.
\finth

A similar ``stabilization'' theorem involving $S_\alpha (E)\cdot
S_\beta (F)$ instead of $S_\lambda (E-F)$ was proved in [5].
Just as in [5] it follows from (17) that the $n=(\mu +1)q$ and
$m=(\mu +1+m-n)q$ case of (17) gives the correct formula for
${_{\phantom{\scriptstyle +l}\mu }^{m+l}G_q^{(n+l)}}(E;F)$
for all integers $l$, when the sets $E$ and $F$ contain
$(n+l)$ and $(m+l)$ variables, respectively.

Before stating the umbral calculus for $\gbm(E;F)$ we need :

\rem Definition $18$|Let $\alpha $, $\beta $ and $\gamma $ be
the three partitions $\alpha =(\alpha _1,\alpha _2,\ldots,\alpha
_n)$, $\beta =(\beta _1,\beta _2,\ldots,\beta _n)$, $\gamma
=(\gamma _1,\gamma _2,\ldots,\gamma _n)$. The sum $\alpha
+\beta $ and direct sum $\alpha \oplus \gamma $ are defined by
  $$\alpha +\beta 
=(\alpha _1+\beta _1,\alpha _2+\beta _2,\ldots,\alpha _n+\beta
_n)\leqno(19{\rm a})$$ 
and
  $$\alpha \oplus \gamma 
=(\alpha _1,\alpha _2,\ldots,\alpha _n,\gamma _1,\gamma
_2,\ldots,\gamma _m).\leqno(19{\rm b})$$ Note that $\alpha
+\beta $ is a partition while $\alpha \oplus \gamma $ may not
be. In general $\alpha \oplus \gamma $ is just a $(n+m)$-tuple.

\rem Definition $20$|Let 
$\phi \equiv (\phi _1,\phi _2,\ldots,\phi _n)$ be an $n$-tuple of
non-negative integers that is not necessarily a partition. If all
the coordinates of $(\phi +\delta _n)
=(\phi _1+n-1,\phi _2+n-2,\ldots,\phi _{n-1}+1,\phi _n)$
are distinct, then there is a unique permutation
$\sigma _\phi \in {\cal S}_n$ that orders the parts of
$(\phi +\delta _n)$ in decreasing order. Denote the resulting 
partition (with distinct parts) by
$\sigma _\phi (\phi +\delta _n)$. In addition, let 
$(\phi )_{\sigma _\phi }$ be the partition
  $$(\phi )_{\sigma _\phi }\equiv 
\sigma _\phi (\phi +\delta _n)-\delta _n.\leqno(21)$$
Now, given any $n$-tuple  
$\phi =(\phi _1,\phi _2,\ldots,\phi _n)$,
we determine $S_\phi (E-F)$ by means of
  $$S_\phi (E-F),\quad{\rm if\ the\ parts\ of}\ 
(\phi +\delta _n)\ {\rm are}\ {\it not}
\ {\rm distinct}.\leqno(22{\rm a})$$
Otherwise,
  $$S_\phi (E-F)=\epsilon (\sigma _\phi )\cdot 
S_{(\phi )_{\sigma \phi }}(E-F),\leqno(22{\rm b})$$
where $(\phi )_{\sigma _\phi }$ is the partition given by (21) 
and $\epsilon (\sigma _\phi )$ is the sign of the permutation
$\sigma _\phi $. (Note that if $\phi $ is a partition, then the
parts of $(\phi +\delta _n)$ are distinct, $\delta _\phi $ is the
identity permutation, and the right-side of (22b) is simply
$(S_\phi (E-F)$.)

\th Theorem 23|One has
  $$S_\lambda (X+Y)=\sum _{\pi ,\,\alpha \subset  \lambda }
c_{\pi \alpha }^\lambda  S_\alpha (Y) S_\pi (X),\leqno(24)$$
where $c_{\pi \alpha }^\lambda $ are the {\rm L--R} coefficients in $(13)$,
and $X$ and $Y$ are two (signed) sums of monomials.
\finth

\th Theorem 25 {\rm (A. Lascoux)}|One has
  $$S_\pi (-1+\gamma _1,\ldots,-1+\gamma _{\mu +1})=
\sum _{\nu \subset  \pi } d_{\pi \nu } S_\nu (A),\leqno(26)$$
where
  $$d_{\pi \nu }=(-1)^{\left|\pi \right|-\left|\nu \right|}
\det\left[{\pi _i+\mu +1-i\choose \nu _j+\mu +1-j}
\right]_{1\le i,\,j\le \mu +1},\leqno(27)$$
and
  $$A=\{\gamma _1,\gamma _2,\ldots,\gamma _{\mu +1}\}.$$
\finth

\pd THEOREM 23 appears in [9 ; see eq. (5.9) on p.~41] and
\pc THEOREM|~25 is due to A. \pd LASCOUX [8] and can be found
in [9 ; see ex. 10, p.~30].

The first step of the umbral calculus is to compute
  $${_\mu ^m G_{q-1}^{(n)}}
(\gamma _1-1,\gamma _2-1,\ldots,\gamma _{\mu
+1}-1,B;F).\leqno(28)$$ 
This is accomplished by means of the
$q-1$ case of \pc THEOREM|~16, and \pc THEOREMS|~23 and 25.
The second (and deepest) part of the umbral calculus is to
replace  $S_\nu (A)\cdot S_\alpha (B-F)$
in the sum giving (28) by
  $$S_{(((\mu +1+m-n)^{\mu +1}+\nu )
\oplus \alpha )}(E-F).\leqno(29)$$
where (29) is determined by the
$\phi =\bigl(((\mu +1+m-n)^{\mu +1}+\nu )\oplus \alpha \bigr)$
case of {\sl Definition}~20. The resulting sum is equal to
$\gbm(E;F)$. That is, we have

\th Theorem 30 {\rm (umbral calculus)}|Let ${_\mu ^m
G_{q-1}^{(n)}}(E;F)$ be given by the $q-1$ case of $(17)$. For
technical reasons assume that the following inequalities hold :
$(\mu +1)q\le n$, $m\ge \mu +1$, $m-n=k$ is constant,
$l(\nu )\le \mu +1$, and $l(\alpha )\le n-(\mu +1)$. We then have
  $$\displaylines{(31)\quad
{_\mu ^mG_q^{(n)}}(E;F)=
(-1)^{\mu +1\choose 2}
\sum _\lambda  a_{\lambda ,q-1}\hfill\cr
\hfill{}\cdot \Biggl\{\sum _{\scriptstyle \pi ,\,\alpha 
\subset  \lambda \atop \scriptstyle \nu \subset  \pi }
c_{\pi \alpha }^\lambda \cdot d_{\pi \nu }\cdot
S_{\bigl(((\mu +1+m-n)^{\mu +1}+\nu )\oplus \alpha \bigr)}(E-F)\Biggr\},\hfill\cr}$$
where $\left|\lambda \right|\le (\mu +1)(\mu +1+k)(q-1)$ and
$\lambda \subset  [(\mu +1+k)(q-1)]^{(\mu +1)(q-1)}$
and where
$S_{\bigl(((\mu +1+m-n)^{\mu +1}+\nu )\oplus \alpha \bigr)}
(E-F)$
is determined by the
$\phi =(((\mu +1+m-n)^{\mu +1}+\nu )\oplus \alpha )$
case of {\sl Definition}~$20$,
$c_{\pi \alpha }^\lambda $ are the {\rm L--R} coefficients
in $(13)$, and the $d_{\pi \nu }$ are defined by $(27)$.\finth

\pd THEOREM 1.22 of [5] expressed ${_\mu ^m G_1^{(n)}}(E;F)$
as a sum of products of Schur functions
$S_\alpha (F)\cdot S_\beta (F)$. Using (15b) to rewrite this sum
immediately gives
  $${_\mu ^m G_1^{(n)}}(E;F)=(-1)^{\mu +1\choose 2}
S_{(\mu +1+m-n)^{\mu +1}}(E-F),\leqno(32)$$
where $(\mu +1+m-n)^{\mu +1}$
denotes the partition consisting of $(\mu +1)$
parts equal to $(\mu +1+m-n)$.

In [5] it took three pages to write
${_1^2 G_2^{(n)}}(E;F)$
as a sum of products of Schur functions.
Starting with (32) and making use of
\pd THEOREM 30 it was shown in [1] that
  $$\displaylines{(33)\quad {_1^2 G_2^{(n)}}(E;F)
=\{S_{2^2}(D)\}^2+
\{-[2S_{4,3}(D)+2S_{2^3,1}(D)\hfill\cr
\quad\qquad\qquad
{}+3S_{3,2^2}(D)+3S_{3^2,1}(D)
+S_{4,3,1}(D)+S_{3,2,1^2}(D)]\hfill\cr
\quad\qquad\qquad
{}+[S_{4,2}(D)+S_{2^2,1^2}(D)+3S_{2^3}(D)
+3S_{3^2}(D)+3S_{3,2,1}(D)]\hfill\cr
\quad\qquad\qquad
{}+[2S_{3,2}(D)+2S_{2^2,1}(D)]+S_{2^2}(D)\},\hfill\cr}$$
where $D$ denotes the set difference $E-F$.

Further applications of \pd THEOREMS 16 and 30,
as well as detailed proofs, can be found in [1].

\vfill\eject
\vglue44pt
\eightpoint
\centerline{REFERENCES}
\nobreak
\vskip 1cm
\article 1|\pd BIEDENHARN (L.C.), \pd GUSTAFSON (R.A.) and
\pd MILNE (S.C.)|An umbral calculus for polynomials
characterizing $U(n)$ tensor operators|Advances in
Math.|51|1984|36--90|
\article 2|\pd BIEDENHARN (L.C.), \pd HOLMAN III (W.) and
\pd MILNE (S.C.)|The invariant polynomials characterizing
U(n)$ tensor operators
\langle p,q,\ldots,q,0,\ldots,0\rangle $
having maximal null space|Advances in Appl.
Math.|1|1980|390--472|
\livre 3|\pd BIEDENHARN (L.C.)
and \pd LOUCK (J.D.)|Angular Momentum in
Quantum Physics : Theory and Applications|Reading, Addison-Wesley
({\sl Encyclopedia of Math. Appl.}, {\bf 8}), $\oldstyle 1981$|
\livre 4|\pd BIEDENHARN (L.C.)
and \pd LOUCK (J.D.)|The Racah-Wigner Algebra in
Quantum Theory|Reading, Addison-Wesley
({\sl Encyclopedia of Math. Appl.}, {\bf 9}), $\oldstyle 1981$|
\article 5|\pd GUSTAFSON (R.A.) and
\pd MILNE (S.C.)|Schur functions and the invariant
characterizing $U(n)$ tensor operators|Advances in
Appl. Math.|4|1983|422--478|
\article 6|\pd GUSTAFSON (R.A.) and
\pd MILNE (S.C.)|Schur functions, Good's identity,
and hypergeometric series well-poised 
in $SU(n)$ tensor operators|Advances in
Math.|48|1983|177--188|
\divers 7|\pd GUSTAFSON (R.A.) and
\pd MILNE (S.C.)|A new symmetry
for Biedenharn's $G$-functions and
classical hypergeometric series,
{\sl Advances in Math.}, in press|
\article 8|\pd LASCOUX (A.)|Classes de Chern d'un produit
tensoriel|C.R. Acad. Sci. Paris|268A|1978|385--387|
\livre 9|\pd MACDONALD (I.G.)|Symmetric Functions and Hall
Polynomials|Oxford Univ. Press, $\oldstyle 1979$|
\article 10|\pd MILNE (S.)|Hypergeometric series well-poised
in $SU(n)$ and a generalization of Biedenharn's
$G$-functions|Advances in Math.|36|1980|169--211|

\tenpoint
\adresse
Stephen C. {\petcap Milne},
Department of Mathematics,
Texan A \& M University,
College Station, Texas 77843, U.S.A.

now at :

Department of Mathematics,
Ohio State University
Columbus, Ohio, U.S.A.

\fin

\bye
