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\auteurcourant={I.G. MACDONALD}
\titrecourant={SCHUR FUNCTIONS : THEME AND VARIATIONS}


\eightpoint
\leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1992$, 498/S--27}
\leftline{Actes 28\emini\ S\'eminaire Lotharingien, p. 5-39}

\tenpoint

\vskip 1.5cm
\centerline{{\bf SCHUR FUNCTIONS : THEME AND VARIATIONS}}
\vskip 2.8mm
\centerline{\sevenrm BY}
\vskip 2.8mm
\centerline{I. G. MACDONALD}
\titre Introduction and theme|
In this article we shall survey various generalizations, analogues
and deformations of Schur functions --- some old, some new ---
that have been proposed at various times. We shall present these as
a sequence of variations on a theme and (unlike e.g. Bourbaki) we
shall proceed from the particular to the general. Thus
Variations~1 and~2 are included in Variation~3 ; Variations~4
and~5 are particular cases of Variation~6 ; and in their turn
Variations~6, 7 and~8 (in part) are included in Variation~9.

To introduce our theme, we recall [${\rm M}_1$, Ch.~I, \S\kern 2pt
3] that the {\it Schur function}
$s_\lambda (x_1,\ldots,x_n)$ (where $x_1$, \dots~, $x_n$ are
independent indeterminates and $\lambda =(\lambda
_1,\ldots,\lambda _n)$ is a partition of ${\rm length}\le n$) may be
defined as the quotient of two alternants :
$$
s_\lambda (x_1,\ldots,x_n)
={\det \bigl(x_i^{\lambda _j+n-j}\bigr)_{1\le i,j\le n}
\over 
\det \bigl(x_i^{n-j}\bigr)_{1\le i,j\le n}}.\leqno(0.1)
 $$
The denominator on the right-hand side is the Vandermonde
determinant, equal to the product
$\prod \limits_{i<j}(x_i-x_j)$.

When $\lambda =(r)$, $s_\lambda $ is the {\it complete symmetric
function}~$h_r$, and when $\lambda =(1^r)$, $s_\lambda $ is the
{\it elementary symmetric function}~$e_r$. In terms of the
$h$'s, the Schur function $s_\lambda $ (in any number of variables)
is given by the Jacobi-Trudi formula
$$
s_\lambda =\det\bigl(h_{\lambda _i-i+j}\bigr)_{1\le i,j\le n}.
\leqno(0.2)
$$

Dually, in terms of the elementary symmetric functions,
$s_\lambda $ is given by the N\"agelsbach-Kostka formula
$$
s_\lambda =\det\bigl(e_{\lambda' _i-i+j}\bigr)_{1\le i,j\le m}
\leqno(0.3)
$$
in which $\lambda '=(\lambda '_1,\ldots,\lambda '_m)$ is the
conjugate [${\rm M}_1$, Ch.~I, \S\kern 2pt 1] of the
partition~$\lambda $.

There are (at least) two other determinantal formulas for
$s_\lambda $ : one in terms of ``hooks" due to Giambelli, and the
other in terms of ``ribbons" discovered quite recently by Lascoux and
Pragacz [${\rm LP}_2$]. If $\lambda =(\alpha _1,\ldots,\alpha
_p\mid \beta _1,\ldots, \beta _p)$ in Frobenius notation
[${\rm M}_1$, Ch.~I, \S\kern 2pt 1], Giambelli's formula is
$$
s_\lambda =\det\bigl(s_{(\alpha _i\mid \beta _j)}\bigr)_{1\le
i,j\le p}.\leqno(0.4) 
$$

To state the formula of Lascoux and Pragacz, let
$$
\lambda ^{(i,j)}=(\alpha _1,\ldots,\widehat \alpha _i,\ldots,
\alpha _p\mid \beta _1,\ldots, \widehat \beta _j,\ldots, \beta _p)
$$
for $1\le i,j\le p$, where the circumflexes indicate deletion of the
symbols they cover ; and let
$$
[\alpha _i\mid \beta _j]=[\alpha _i\mid \beta _j]_\lambda
=\lambda -\lambda ^{(i,j)}.
 $$
In particular, $[\alpha _1\mid \beta _1]$ is the {\it rim} or {\it
border} of $\lambda $, and $[\alpha _i\mid \beta _j]$ is that part of
the border consisting of the squares $(h,k)$ such that $h\ge i$ and
$k\ge j$. With this notation explained, the ``ribbon formula" is
$$
s_\lambda =\det\bigl(s_{[\alpha _i\mid \beta _j]}\bigr)_{1\le
i,j\le p}.\leqno(0.5) $$

Finally, we recall [${\rm M}_1$, Ch.~I, \S\kern 2pt 5] the expression
of a Schur function as a sum of monomials : namely
$$
s_\lambda =\sum _T x^T \leqno(0.6)
$$
summed over all column-strict tableaux $T$ of shape $\lambda $,
where
$x^T=\prod \limits_{s\in \lambda } x_{T(s)}$. (Throughout this
article, we shall find it convenient to think of a tableau~$T$ as a
mapping from (the shape of) $\lambda $ into the positive integers,
so that $T(s)$ is the integer occupying the square $s\in \lambda $.)

All these formulas, with the exception of the original definition
(0.1), have their extensions to skew Schur functions 
$s_{\lambda/\mu }$. 
In place of (0.2) we have
$$\leqalignno{
s_{\lambda/\mu } &=\det\bigl(h_{\lambda _i-\mu _j-i+j}\bigr),&
(0.7) \cr
\noalign{\medskip}
\noalign{\hbox{and in place of (0.3) we have}}
\noalign{\medskip}
s_{\lambda/\mu } 
&=\det\bigl(e_{\lambda' _i-\mu'_j-i+j}\bigr)&(0.8)\cr
}$$
where $\lambda '$, $\mu '$ are the partitions conjugate to $\lambda
$, $\mu $ respectively. For the skew versions of (0.4) and (0.5) we
refer to $[{\rm LP}_1]$, 
$[{\rm LP}_2]$. Finally, in place of (0.6) we have
$$
s_{\lambda /\mu }=\sum _T x^T\leqno(0.9)
$$
where now $T$ runs over column-strict tableaux of shape
$\lambda -\mu $
[${\rm M}_1$, Ch.~I, \S\kern 2pt 5].

To complete this introduction we should mention the {\it Cauchy
identity}
$$\leqalignno{
\prod _{i,j}(1-x_iy_j)^{-1}&=\sum _\lambda s_\lambda (x)
s_\lambda (y)&(0.10)\cr
\noalign{\hbox{and its dual version}}
\noalign{\medskip}
\prod _{i,j}(1+x_iy_j)&=\sum _\lambda s_\lambda (x)
s_{\lambda'} (y)&(0.11)\cr
}
$$
where $\lambda '$ is the conjugate of $\lambda $.

If we replace each $y_j$ by $y_j^{-1}$ and then multiply by a
suitable power of $y_1y_2\ldots$, (0.11) takes the equivalent form
(when the number of variables $x_i$, $y_j$ is finite)
$$
\prod _{\hbox{\vrule width0pt height0pt depth 4pt}\scriptstyle
1\le i\le n\atop \scriptstyle 1\le j\le m} (x_i+y_j)=\sum _\lambda 
s_\lambda (x) s_{\,\widehat \lambda '}(y) 
\leqno(0.11')$$
summed over partitions 
$\lambda =(\lambda _1,\ldots, \lambda_n)$ 
such that $\lambda _1\le m$, where 
$\widehat \lambda=(\widehat \lambda _1,\ldots ,\widehat
\lambda _n)$ 
is the complementary partition defined by 
$\widehat\lambda _i=m-\lambda _{n+1-i}$, and $\widehat
\lambda  '$ is the conjugate of~$\widehat\lambda $.

The left-hand side of (0.10) may be regarded as defining a scalar
product $\langle f,g\rangle$ on the ring of symmetric functions, as
follows. For each $r\ge 1$ let $p_r$ denote the $r\,$th power sum 
$\sum x_i^r$, and for each partition 
$\lambda =(\lambda _1,\lambda _2,\ldots)$ 
let $p_\lambda $ denote the product
$p_{\lambda _1}p_{\lambda _2}\ldots$
The $p_\lambda $ form a $\bboard Q$-basis of the ring of
symmetric functions (in infinitely many variables, 
\cf. [${\rm M}_1$, Ch.~I]) with rational coefficients, and the scalar
product may be defined by
$$
\langle p_\lambda ,p_\mu \rangle=\delta _{\lambda \mu }
z_\lambda \leqno(0.12)
 $$
where $\delta _{\lambda \mu }$ is the Kronecker delta, and
$$
z_\lambda =\prod _{i\ge 1} i^{m_i}.m_i!,
$$
$m_i=m_i(\lambda )$ being the number of parts $\lambda _j$ of
$\lambda $ equal to~$i$, for each~$i\ge 1$.

The Cauchy formula (0.10) is now equivalent to the statement that
the Schur functions $s_\lambda $ form an orthonormal basis of the
ring of symmetric functions, i.e.,
$$
\langle s_\lambda ,s_\mu \rangle=\delta _{\lambda \mu }.
\leqno(0.13)
$$
Also, from this point of view, the skew Schur function
$s_{\lambda /\mu }$ may be defined to be 
$s_\mu ^{\perp}(s_\lambda )$,
where $s_\mu ^{\perp}$ is the adjoint of multiplication by
$s_\mu $, so that
$\langle s_\mu ^{\perp}f,g\rangle=\langle f,s_\mu g\rangle$
for any symmetric functions $f$, $g$.

\titre 1st Variation : Hall-Littlewood symmetric functions|
Let $x_1$, \dots~, $x_n$, $t$ be independent variables and let
$\lambda =(\lambda _1,\ldots,\lambda _n)$ be a partition of 
${\rm length} \le n$. The {\it Hall-Littlewood symmetric function}
indexed by~$\lambda $ 
[${\rm M}_1$, Ch.~III]
is defined by
$$
P_\lambda (x_1,\ldots, x_n;t)={1\over v_\lambda (t)}
\sum _{w\in {\goth S}_n}
w\Bigl(x_1^{\lambda _1}\ldots x_n^{\lambda _n}
\prod _{i<j} {x_i-tx_j\over x_i-x_j}\Bigr)\leqno(1.1)
$$
in which $v_\lambda (t)\in {\bboard Z}[t]$ is a polynomial (with
constant term equal to~1) chosen so that the leading monomial in
$P_\lambda $ is 
$x^\lambda =x_1^{\lambda  _1}\ldots x_n^{\lambda _n}$. 
When $t=0$, the right-hand side of (1.1) is just the expansion
of the determinant 
$\det(x_i^{\lambda _j+n-j})$, divided by the Vandermonde
determinant, so that when $t=0$ the formula (1.1) reduces to the
definition (0.1) of the Schur function.

None of determinantal formulas (0.2) -- (0.5) have counterparts for
the Hall-Littlewood functions (so far as I am aware). In place of
(0.6) we have
$$
P_\lambda (x;t)=\sum _T \psi _T(t) x^T \leqno(1.2)
$$
summed over column-strict tableaux $T$ of shape~$\lambda $,
where
$\psi _T(t)\in {\bboard Z}[t]$ is a polynomial given explicitly in
[${\rm M}_1$, Ch.~III, \S\kern2pt 5].

Finally, in place of the Cauchy identity (0.10) we have
$$
\prod _{i,j} {1-tx_iy_j \over 1-x_iy_j}
=\sum _\lambda  b_\lambda (t) P_\lambda (x;t) P_\lambda (y;t).
\leqno(1.3) 
$$As in the case of the Schur functions, this identity may be
interpreted as saying that the symmetric functions $P_\lambda
(x;t)$ are pairwise orthogonal with respect to the scalar product
defined in terms of the power-sum products~by
$$
\langle p_\lambda ,p_\mu \rangle_{_{\scriptstyle t}}
=\delta _{\lambda \mu } z_\lambda  
\prod _{i\ge 1} (1-t^{\lambda _i})^{-1}. \leqno(1.4)
$$
For more details, and in particular for the definition of the
polynomials $b_\lambda (t)$ featuring in the right-hand side of
(1.3), we refer to [${\rm M}_1$, Ch.~III].

\goodbreak

\titre 2nd Variation : Jack symmetric functions|
These are symmetric functions $P_\lambda ^{(\alpha )}(x)$
depending on a parameter~$\alpha $, but unlike the Hall-Littlewood
functions (Variation~1) there is no closed formula such as (1.1) that
can serve as definition. The simplest (and original) definition is the
following : analogously to (0.12) and (1.4), we define a scalar
product~by
$$ \langle p_\lambda ,p_\mu
\rangle^{(\alpha )}=\delta _{\lambda \mu } z_\lambda  
\alpha ^{l(\lambda )}\leqno(2.1) 
$$
where $l(\lambda )$ is the length of the partition $\lambda $, that
is to say the number of non zero parts~$\lambda _i$. For each
positive integer~$n$, arrange the partitions of~$n$ in
lexicographical order (so that $(1^n)$ comes first and $(n)$ comes
last). Then the $P_\lambda ^{(\alpha )}(x)$ are uniquely determined
by the two requirements
$$
P_\lambda ^{(\alpha )}(x)=x^\lambda +{\rm lower\ terms}
\leqno(2.2)
$$
where $x^\lambda $ denotes the monomial $x_1^{\lambda _1}
x_2^{\lambda _2}\ldots$, and by ``lower terms" is meant a sum of
monomials $x^\beta $ corresponding to sequences
$\beta =(\beta  _1,\beta  _2,\ldots\,)$ that precede~$\lambda $ in
the lexicographical order ; and
$$ \langle P_\lambda^{(\alpha )} ,P_\mu^{(\alpha )}
\rangle^{(\alpha )}=0\qquad {\rm if}\ \lambda \not=\mu. \leqno(2.3) 
$$
The two conditions mean that the $P_\lambda ^{(\alpha )}$ may be
constructed from the monomial symmetric functions by the
Gram-Schmidt process, starting (for partitions of~$n$) with
$P_{(1^n)}=e_n$, the $n\,$th elementary symmetric function.

Since the scalar product (2.1) reduces to (0.12) when $\alpha =1$, it
follows that 
$P_\lambda ^{(\alpha )}=s_\lambda $ when $\alpha =1$.

In view of the definition (2.1) of the scalar product, the
orthogonality property (2.3) is equivalent to the following
generalization of the Cauchy identity (0.10) :
$$
\prod _{i,j} (1-x_iy_j)^{-1/\alpha }=\sum _\lambda 
c_\lambda(\alpha )  
P_\lambda ^{(\alpha )}(x) P_\lambda ^{(\alpha)}(y) \leqno(2.4)
 $$
where the $c_\lambda (\alpha )$ are rational functions of the
parameter~$\alpha $ which have been calculated explicitly by
Stanley~[S] --- note, however, that his normalization of the Jack
symmetric functions is different from ours.

As in the case of the Hall-Littlewood symmetric functions, none of
the determinantal formulas (0.2) -- (0.5) generalize, so far as is
known, to the present situation. In place of (0.6) there is an explicit
expression for $P_\lambda ^{(\alpha )}(x)$ as a weighted sum of
monomials, namely
$$
P_\lambda ^{(\alpha )}(x)=\sum _T f_T(\alpha ) x^ T \leqno(2.5)
$$
summed over column-strict tableaux $T$ of shape~$\lambda $,
where $f_T(\alpha )$ is a rational function of~$\alpha $, computed
explicitly by Stanley~[S], to whom we refer for more details.

Finally, the dual Cauchy formula (0.11) generalizes as follows :
$$
\prod _{i,j} (1+x_iy_j)=\sum _\lambda 
P_\lambda ^{(\alpha )}(x) P_{\lambda '}^{(1/\alpha )}(y)
\leqno(2.6)
$$
where as before $\lambda '$ is the conjugate of $\lambda $.

\titre 3rd Variation|
Our third variation is a family of symmetric functions 
$P_\lambda(x;q,t)$, indexed as usual by partitions~$\lambda$, and
depending on two parameters~$q$ and~$t$.
They include the two previous variations (the Hall-Littlewood
symmetric functions and the Jack symmetric functions) as
particular cases (see below). Since I have given an extended account
of these symmetric functions at a previous S\'eminaire
Lotharingien~[${\rm M}_2$], I shall be brief here and refer to 
{\it loc. cit.} for all details. The functions may be most simply
defined along the same lines as in Variation~2 : we define a new
scalar product on the ring of symmetric functions~by
$$
\langle p_\lambda , p_\mu \rangle_{_{\scriptstyle q,t}}
=\delta _{\lambda ,\mu } z_\lambda  
\prod _{i\ge 1} {1-q^{\lambda _i} \over 1-t^{\lambda _i}},
\leqno(3.1)
$$
and then the symmetric functions
$P_\lambda (x;q,t)$ are uniquely determined by the two
requirements
$$
\displaylines{
\rlap{(3.2)}\hfill
P_\lambda (x;q,t)=x^\lambda +{\rm lower\ terms},\hfill\cr
\rlap{(3.3)}\hfill
\langle P_\lambda ,P_\mu \rangle_{_{\scriptstyle q,t}} =0
\qquad {\rm if}\ \lambda \not=\mu .\hfill\cr}
$$

If we set $q=t^\alpha $ and then let $t\rightarrow 1$, in the limit
the scalar product (3.1) becomes that defined in (2.1), from which it
follows that the Jack symmetric function 
$P_\lambda ^{(\alpha)}(x)$ is the limit of 
$P_\lambda (x;t^\alpha ,t)$ as $t\rightarrow 1$.
Again, if we set $q=0$ the scalar product (3.1) reduces to (1.4), and
it follows that $P_\lambda (x;0,t)$ is the Hall-Littlewood
symmetric function $P_\lambda (x;t)$. Finally, if $q=t$ then (3.1)
reduces to the original scalar product (0.12), and correspondingly 
$P_\lambda (x;q,q)$ is the Schur function~$s_\lambda (x)$.

In view of the definition (3.1) of the scalar product, the
orthogonality condition (3.3) is equivalent to the following
extension of the Cauchy identity (0.10) :
$$
\prod _{i,j} {(tx_iy_j;q)_\infty \over(x_iy_j;q)_\infty }
=\sum _\lambda b_\lambda (q,t)
P_\lambda (x;q,t) P_\lambda (y;q,t).\leqno(3.4)
$$
On the left-hand side of (3.4) we have used the standard notation
$$
(x;q)_\infty =\prod _{i\ge 0} (1-xq^i).
$$
On the right-hand side, $b_\lambda (q,t)$ is a rational function of
$q$ and $t$, given explicitly in
[${\rm M}_2$, \S\kern 2pt 5].

As in the previous two variations, none of the determinantal
formulas for Schur functions quoted in the introduction appear to
generalize to the present situation. However, the formula (0.6) for
$s_\lambda $ as a sum of monomials does generalize : namely we
have
$$
P_\lambda (x;q,t)=\sum _T \varphi_{_{\scriptstyle T}} (q,t) x^T
\leqno(3.5) 
$$
where $\varphi_{_{\scriptstyle T}} (q,t)$ 
is a rational function of $q$ and $t$, again
given explicit expression in
[${\rm M}_2$, \S\kern 2pt 5].

Finally, the dual Cauchy formula (0.11) generalizes as follows
[${\rm M}_2$, \S\kern 2pt 5] :
$$
\prod _{i,j}(1+x_iy_j)=\sum _\lambda P_\lambda (x;q,t)
P_{\lambda'} (y;t,q).\leqno(3.6)
$$

\titre 4\thinspace th Variation : factorial Schur functions|
Let $z=(z_1,\ldots, z_n)$ be a sequence of independent variables.
For each pair of partitions $\lambda $, $\mu $ Biedenharn and Louck
have defined a {\it skew factorial Schur function}
$t_{\lambda /\mu }(z)$ in [${\rm BL}_1$]. Their original definition 
({\it loc. cit.}) was couched in terms of Gelfand patterns, and in the
equivalent language of tableaux it reads as follows. If 
$T:\lambda -\mu \rightarrow [1,n]$
is a column-strict tableau of shape $\lambda -\mu $, containing
only the integers $1$, $2$, \dots~,~$n$, let
$$
z^{(T)}=\prod _{s\in \lambda -\mu } \bigl(z_{T(s)}
-T^*(s)+1\bigr),\leqno(4.1)
$$
where $T^*(i,j)=T(i,j)+j-i$
(so that $T^*$ is a {\it row-strict} tableau of shape 
$\lambda -\mu $). Then $t_{\lambda /\mu }(z)$ is defined~by
$$
t_{\lambda /\mu }(z)=\sum _T z^{(T)} \leqno(4.2)
$$
summed over all column-strict tableaux
$T:\lambda -\mu \rightarrow [1,n]$.

When $\mu =0$ they write $t_\lambda $ in place of 
$t_{\lambda /0}$.

It is not particularly obvious from this definition that $t_{\lambda
/\mu }(z)$ is in fact a (non-homogeneous) {\it symmetric}
polynomial in
$z_1$,  \dots~,~$z_n$, and Biedenharn and Louck had some trouble
(see  [${\rm BL}_1$] pp.~407--412) in establishing this fact
directly from their definition~(4.2).

Some time ago I noticed that it followed rather simply from one of
their results
(Th.~5 of  [${\rm BL}_2$]) that an alternative definition of
$t_\lambda (z)$ could be given which brought out its analogy with
the Schur function~$s_\lambda $ : namely (for
$\lambda =(\lambda _1,\ldots,\lambda _n)$ a partition of 
${\rm length}\le n$)
$$\displaylines{
\rlap{(4.3)}\hfill
t_\lambda (z)=\det \bigl(z_i^{(\lambda _j+n-j)}\bigr)
\bigm/\det\bigl(z_i^{(n-j)}\bigr),\hfill\cr
\noalign{\hbox{where $z^{(r)}$ is the ``falling factorial"}}
\noalign{\medskip}
\rlap{(4.4)}\hfill
z^{(r)}=z(z-1)\ldots (z-r+1)\qquad (r\ge 0).\hfill\cr}
$$
Note that since $z^{(r)}$ is a monic polynomial in $z$ of degree~$r$,
the denominator in (4.3) is just the Vandermonde determinant :
$$
\det\bigl(z_i^{(n-j)}\bigr)=\det\bigl(z_i^{n-j}\bigr)
=\prod _{i<j} (z_i-z_j).
$$
Hence $t_\lambda $ as defined by (4.3) is the quotient of a
skew-symmetric polynomial in $z_1$, \dots~,~$z_n$ by the
Vandermonde determinant, and is therefore a (non-homogeneous)
symmetric polynomial in the~$z_i$. Moreover, it is clear from (4.3)
that $t_\lambda (z)$ is 
of the form
$$
t_\lambda (z)=s_\lambda (z)+{\rm terms\ of\ lower\ degree},
$$
and hence that the $t_\lambda (z)$, as $\lambda $ runs through the
partitions of ${\rm length} \le n$, form a ${\bboard Z}$-basis of the
ring $\Lambda _n$ of symmetric polynomials in
$z_1$, \dots~,~$z_n$.

In [CL], Chen \& Louck show that $t_\lambda $ (and more generally
$t_{\lambda /\mu }$) satisfies a determinantal identity analogous
to (0.2) and (0.7). Namely if
$$
w_r(z)=t_{(r)}(z)
$$
for all $r\ge 0$ (and $w_r(z)=0$ when $r<0$) then we have ({\it loc.
cit.}, Th.~5.1)
$$
t_{\lambda /\mu }(z)=\det\bigl(w _{\lambda _i-\mu _j-i+j}
(z-\mu _j+j-1)\bigr)\leqno(4.5)
$$
where in general $z+r$ denotes the sequence
$(z_1+r,\ldots, z_n+r)$.

The other determinantal formulas quoted in the introduction all
have their analogues for factorial Schur functions. If we define
$$
f_r(z)=t_{(1^r)}(z)\qquad (0\le r\le n)
$$
(and $f_r(z)=0$ for $r<0$ and $r>n$), so that the $f_r$ are the
analogues of the elementary symmetric functions, then we have
$$
t_{\lambda /\mu }(z)=\det
\bigl(f_{\lambda '_i-\mu _j'-i+j}(z+\mu _j'-j+1)\bigr).
\leqno(4.6)
$$

We shall not stop to prove (4.6) here, nor the hook and ribbon
formulas
$$
\leqalignno{t_\lambda (z)
&=\det\bigl(t_{(\alpha _i\mid\beta _j)}(z)\bigr)_{1\le i,j\le r}
&(4.7)\cr
&=\det\bigl(t_{[\alpha _i\barre \beta _j]}(z)\bigr)_{1\le i,j\le r}\cr}
$$
(where 
$\lambda =(\alpha _1,\ldots, \alpha _r\barre \beta _1,\ldots,
\beta _r)$ 
in Frobenius notation, and for the explanation of the
notation $[\alpha _i\barre \beta _j]$ we refer to (0.5)), since they
are special cases of the corresponding results in Variation~6, which
in their turn are contained in Variation~9. In this development we
take (4.3) and (4.5) as definitions of $t_\lambda $ and  
$t_{\lambda /\mu}$ respectively, and deduce (4.2) from them (see
(6.16) below), very much in the spirit of [${\rm M}_1$], Chapter~I,
\S\kern2pt 5.

\def\smashbot#1{\setbox0=\hbox{#1}\dp0=0pt\box0}

\titre 5\thinspace th Variation : $\alpha $-paired factorial Schur
functions| 
Let $z=(z_1,\ldots, z_n)$ again be a sequence of
independent variables, and let $\alpha $ be another variable (or
parameter). In parallel with the factorial Schur functions
(Variation~4) Biedenharn and Louck [${\rm BL}_1$] have defined
$\alpha $-{\it paired factorial Schur functions} $T_{\lambda /\mu
}(\alpha ;z)$. As in the previous case, their definition was couched in
terms of Gelfand patterns, and in the equivalent language of tableaux
it reads as follows.
Let
$$
\overline z_i=-\alpha -z_i\qquad (1\le i\le n)
$$
and for each column-strict tableau
$T:\lambda -\mu \rightarrow [1,n]$ let
$$
(\alpha :z)^{(T)}=\prod _{s\in \lambda -\mu }
\bigl(z_{T(s)}-T^*(s)+1\bigr)\bigl(\overline z_{T(s)}
-T^*(s)+1\bigr)\leqno(5.1)
$$
where (as in \S\kern2pt 4) $T^*$ is the row-strict tableau
associated with~$T$ (i.e., $T^*(i,j)=T(i,j)+j-i$). Then
$$
T_{\lambda /\mu }(\alpha ;z)=\sum _T(\alpha :z)^{(T)}
\leqno(5.2)
$$
summed over all column-strict tableaux $T:\lambda -\mu
\rightarrow [1,n]$.
(When $\mu =0$, they write $T_\lambda $ in place of 
$T_{\lambda  /0}$.)

Chen and Louck remark ([CL], p.~18) that ``it is quite surprising that
the $\alpha $-paired factorial Schur function enjoys all the
properties of the ordinary factorial Schur function." The reason for
this, we believe, lies in the fact that both these classes of
symmetric functions are special cases of those to be defined in our
$6\,$th Variation. In the present situation the falling factorial 
$z^{(r)}$
is replaced~by
$$\displaylines{
z^{(r)}\,\overline z^{\kern2pt(r)}
=\prod _{i=0}^{r-1} (z-i)(\overline z-i)\cr
\noalign{\hbox{where $\overline z=-\alpha -z$ ; and since}}
(z-i)(\overline z-i)=z\overline z +\alpha i+i^2\cr
\noalign{\hbox{it follows that we may write}}
z^{(r)}\,\overline z^{\kern2pt(r)}=\prod _{i=1}^r (x+a_i)\cr}
$$
where $x=z\overline z$ and $a_i=\alpha (i-1)+(i-1)^2$. In
Variation~6 below the building blocks are the products
\smashbot{$(x\barre a)^r=\prod \limits_{i=1}^r (x+a_i)$}\  
defined by
an {\it arbitrary} sequence $a_1$, $a_2$, \dots

We may then take as an alternative definition of 
$T_\lambda (\alpha ;z)$, where $\lambda $ is a partition of 
${\rm length}\le n$, 
$$
T_\lambda (\alpha ;z)={\det\bigl(z_i^{(\lambda _j+n-j)}\,
\overline z_i^{\kern2pt(\lambda _j+n-j)}\bigr)
\over \det\bigl(z_i^{(n-j)}\, \overline z_i^{\kern2pt(n-j)}\bigr)}
\leqno(5.3)
$$
([CL], Th. 6.2) ; all the determinantal formulas (Jacobi-Trudi etc.)
together with the tableau definition (5.2) are consequences of (5.3),
as we shall show in a more general context in the next section.

\titre 6\thinspace th Variation|
Let $R$ be any commutative ring and let $a=(a_n)_{n\in {\bboard Z}}$
be any (doubly infinite) sequence of elements of~$R$. For each
$r\in {\bboard Z}$ we define $\tau ^r a$ to be the sequence whose
$n$th term is $a_{n+r}$ :
$$\displaylines{
(\tau ^ra)_n=a_{n+r}.\cr
\noalign{\hbox{\indent Let}}
(x\barre a)^r=(x+a_1)\ldots (x+a_r)\cr
\noalign{\hbox{for each $r\ge 0$. Clearly we have}}
\noalign{\medskip }
\rlap{(6.1)}\hfill
(x\barre a)^{r+s}=(x\barre a)^r \,(x\barre \tau ^ra)^s\hfill\cr
}
$$
for all $r,\,s\ge 0$.

\goodbreak
Now let $x=(x_1, \dots, x_n)$ be a sequence of independent
indeterminates over~$R$, and for each 
$\alpha =(\alpha _1,\ldots, \alpha _r)\in {\bboard N}^n$ define
$$
A_\alpha (x\barre a)
=\det \bigl((x_i\barre a)^{\alpha _j}\bigr)_{1\le i,j\le n}.
\leqno(6.2) 
$$
In particular, when $\alpha =\delta =(n-1,n-2,\ldots, 1,0)$, since
$(x_i\barre a)^{n-j}$ 
is a monic polynomial in~$x_i$ of degree~$(n-j)$,
it follows that
$$
A_\delta (x\barre a)=\det\bigl(x_i^{n-j}\bigr)=\prod _{i<j}(x_i-x_j)
\leqno(6.3)
$$
is the Vandermonde determinant $\Delta (x)$, independent of the
sequence~$a$. Since $A_\alpha (x\barre a)$ is a skew symmetric
polynomial in
$x_1$, \dots~, $x_n$, it is therefore divisible by
$A_\delta (x\barre a)$ in $R[x_1,\ldots, x_n]$. Moreover, the
determinant $A_\alpha (x\barre a)$ clearly vanishes if any two of the
$\alpha _i$  are equal, and hence (up to sign) we may assume that
$\alpha _1>\cdots>\alpha _n\ge 0$, i.e.,
that $\alpha =\lambda +\delta $ where 
$\lambda =(\lambda_1,\ldots, \lambda _n)$ is a partition of 
${\rm length}\le n$. It follows therefore that
$$
s_\lambda (x\barre a)=A_{\lambda +\delta }(x\barre a)
\bigm/ A_\delta (x\barre a)\leqno(6.4)
$$
is a symmetric (but not homogeneous) polynomial in
$x_1$, \dots~, $x_n$ with coefficients in~$R$. Moreover it is
clear from the definitions that
$$
A_{\lambda +\delta }(x\barre a)=a_{\lambda +\delta }(x)+
{\rm lower\ terms},
$$
in the notation of $[{\rm M}_1]$, ch.~I, and hence that
$$
s_\lambda (x\barre a)=s_\lambda (x)+
{\rm terms\ of\ lower\ degree.}
$$
Hence the $s_\lambda (x\barre a)$ form an $R$-basis of the ring
$\Lambda _{n,R}=R[x_1,\ldots,x_n]^{{\goth S}_n}$.

These polynomials $s_\lambda (x\barre a)$, and their skew analogues
$s_{\lambda /\mu }(x\barre a)$ to be defined later, form our $6$th
Variation. They include Variations~4 and~5 as special cases : for
Variation~4 we take
$R={\bboard Z}$, $x_i=z_i$ and $a_n=1-n$ 
for all $n\in {\bboard Z}$ ; for Variation~5 we take
$R={\bboard Z}[\alpha ]$, $x_i=z_i\overline z_i$ and
$a_n=(n-1)\alpha +(n-1)^2$. The Schur functions themselves are
given by the zero sequence :
$a_n=0$ for all $n\in {\bboard Z}$. When $\lambda =(r)$ we shall 
write
$$
h_r(x\barre a)=s_{(r)}(x\barre a)\qquad (r\ge 0)
$$
with the usual convention that $h_r(x\barre a)=0$ if $r<0$ ; and when
$\lambda =(1^r)$ $(0\le r\le n)$ we shall write
$$
e_r(x\barre a)=s_{(1^r)}(x\barre a)
\qquad (0\le r\le n)
$$
with the convention that $e_r(x\barre a)=0$ if $r<0$ or $r>n$.

\goodbreak
Let $t$ be another indeterminate and let
$$
f(t)=\prod _{i=1}^n(t-x_i).
$$
From (6.3) it follows that
$$
f(t)=A_{\delta _{n+1}}(t,x_1,\ldots,x_n\barre a)
\bigm/ A_{\delta _n}(x_1,\ldots, x_n\barre a).
$$
By expanding the determinant 
$A_{\delta _{n+1}}$ along the top row we shall obtain
$$
f(t)=\sum _{r=0}^n (-1)^r e_r(x\barre a)(t\barre a)^{n-r}.\leqno(6.5)
$$

Let ${\bf E}(x\barre a)$, ${\bf H}(x\barre a)$ be the (infinite) matrices
$$
\eqalign{{\bf H}(x\barre a)
&=\bigl(h_{j-i}(x\barre \tau ^{i+1}a)\bigr)_{i,j\in {\bboard Z}},\cr
{\bf E}(x\barre a)
&=\bigl((-1)^{j-i}
e_{j-i}(x\barre \tau ^{j}a)\bigr)_{i,j\in {\bboard Z}}.\cr 
}
$$
Both are upper unitriangular, and they are related by
$$
{\bf E}(x\barre a)={\bf H}(x\barre a)^{-1}.\leqno(6.6)
$$

{\it Proof}\pointir
We have to show that
$$
\sum _j (-1)^{k-j} e_{k-j}(x\barre \tau ^ka)\,
h_{j-i}(x\barre \tau ^{i+1}a)=\delta _{ik}
$$
for all $i,k$. This is clear if $i\ge k$, so we may assume $i<k$.
Since $f(x_i)=0$ it follows from (6.5) that
$$
\sum _{r=0}^n (-1)^r e_r(x\barre a)\,(x_i\barre a)^{n-r}=0
$$
and hence, replacing $a$ by $\tau ^{s-1}a$ and multiplying by
$(x_i\barre a)^{s-1}$, that
$$
\sum _{r=0}^n (-1)^r e_r(x\barre \tau ^{s-1}a)\,
(x_i\barre a)^{n-r+s-1}=0\leqno(1)
$$
for all $s> 0$ and $1\le i\le n$.
Now it is clear, from expanding the determinant
$A_{(m)+\delta }(x\barre a)$ down the first column, that
$h_m(x\barre a)$ is of the form
$$
h_m(x\barre a)=\sum _{i=1}^n (x_i\barre a)^{m+n-1}u_i(x)
\leqno(2)
$$
with coefficients $u_i(x)$ rational functions of~$x_1$, 
\dots~, $x_n$ independent
of~$m$. (In fact, it is easily seen that $u_i(x)=1/f'(x_i)$.)

From (1) and (2) it follows that
$$
\sum _{r=0}^n (-1)^r e_r(x\barre \tau ^{s-1}a)\,h_{s-r}(x\barre a)=0
$$
for each $s>0$. Putting $s=k-i$ and replacing $a$ by
$\tau ^{i+1}a$ we obtain
$$
\sum _{i\le j\le k}(-1)^{k-j}
e_{k-j}(x\barre \tau  ^ka)\,h_{j-i}(x\barre \tau ^{i+1}a)=0,
$$
as required.\cqfd

\goodbreak
Next, we have analogues of the Jacobi-Trudi and
N\"agelsbach-Kostka formulas (0.2), (0.3) :

\par\medskip
\noindent
(6.7) {\it If $\lambda $ is a partition of ${\it length}\le n$, then
$$
\eqalign{s_\lambda (x\barre a)&=\det\bigl(h_{\lambda _i-i+j}(x\mid
\tau ^{1-j}a)\bigr)\cr
&=\det\bigl(e_{\lambda '_i-i+j}(x\barre \tau ^{j-1}a)\bigr).\cr}
$$}

\medskip
{\it Proof}\pointir
Let $\alpha =(\alpha _1,\ldots,\alpha _n)\in {\bboard N}^n$. From
equation~(2) above we have
$$\eqalign{
h_{\alpha _i-n+j}(x\barre \tau ^{1-j}a)
&=\sum _{k=1}^n (x_k\barre \tau ^{1-j}a)^{\alpha _i+j-1} u_k(x)\cr
&=\sum _{k=1}^n (x_k\barre a)^{\alpha _i}\,
(x_k\barre \tau ^{1-j}a)^{j-1} u_k(x)\cr} $$
by (6.1). This shows that the matrix
$H_\alpha =\bigl(h_{\alpha _i-n+j}(x\barre \tau ^{1-j}a)\bigr)_{i,j}$
is the product of the matrices 
$\bigl((x_k\barre a)^{\alpha _i}\bigr)_{i,k}$ and 
$B=\bigl((x_k\barre \tau ^{1-j}a)^{j-1}
u_k(x)\bigr)_{k,j}$.
On taking determinants it follows that
$$
\det(H_\alpha )=A_\alpha \,\det(B).
$$
In particular, when $\alpha =\delta $, the matrix
$H_\delta =\bigl(h_{j-i}(x\barre \tau ^{1-j}a)\bigr)$ is unitriangular
and hence has determinant equal to~1. It follows that
$A_\delta \,\det(B)=1$ and hence that
$$
\det(H_\alpha )=A_\alpha (x\barre a)\bigm/A_\delta (x\barre a),
$$
for all $\alpha \in {\bboard N}^n$. Taking 
$\alpha =\lambda +\delta $, we obtain the first of the
formulas~(6.7). The second formula, involving the~$e$'s, is then
deduced from it and (6.6), exactly as in the case of Schur functions
($[{\rm M}_1]$, ch.~I, (2.9)).\cqfd

\rem Remark|A consequence of (6.7) is that the determinant
$$\det\bigl(h_{\lambda _i-i+j}(x\barre \tau ^{1-j}a)\bigr),$$
which appears to involve not only $a_1$, $a_2$, \dots\ but
also $a_0$, $a_{-1}$, \dots~, $a_{2-l(\lambda )}$, is in fact
independent of the latter.

\medskip

More generally, if $\lambda $ and $\mu $ are partitions we define
$$
\leqalignno{
s_{\lambda /\mu }(x\barre a)
&=\det\bigl(h_{\lambda _i-\mu _j-i+j}
(x\barre \tau ^{\mu_j-j+1}a)\bigr) &(6.8)\cr
\noalign{\medskip}
\noalign{\hbox{and then it follows as above from (6.6) that}}
\noalign{\medskip}
s_{\lambda /\mu }(x\barre a)
&=\det\bigl(e_{\lambda _i'-\mu _j'-i+j}
(x\barre \tau ^{-\mu _j'+j-1}a)\bigr).&(6.9)\cr
}
$$

Moreover,
$$
s_{\lambda /\mu }(x\barre a)=0
\ {\it unless}\ 0\le \lambda _i'-\mu _i' \le n
\ {\it for\ all}\ i.\leqno(6.10)
$$

The proof is the same as for Schur functions :
$[{\rm M}_1]$ ch.~I, \S\kern2pt 5.

\medskip
The hook and ribbon formulas (0.4), (0.5) remain valid in the present
context : if
$\lambda 
=(\alpha _1,\ldots,\alpha _p\barre \beta _1,\ldots, \beta_p)$ in
Frobenius notation, then
$$
\leqalignno{
s_\lambda (x\barre a)&=\det\bigl(s_{(\alpha _i\barre \beta _j)}
(x\barre a)\bigr)_{1\le i,j\le p}&(6.11)\cr
&=\det\bigl(s_{[\alpha _i\barre \beta _j]}
(x\barre a)\bigr)_{1\le i,j\le p}.\cr}
$$
This will be considered in a more general context in \S\kern2pt 9.

\par\medskip
Let $y=(y_1,\ldots, y_m)$ be another set of indeterminates, and let
$(x,y)$ denote $(x_1,\ldots,x_n,y_1,\ldots,y_m)$. Then we have
 $$
\leqalignno{
{\bf E}(x,y\barre a)&={\bf E}(y\barre \tau ^na)\,{\bf E}(x\barre a),
&(6.12)\quad {\rm (i)}\cr
{\bf H}(x,y\barre a)&={\bf H}(x\barre a)\,{\bf H}(y\barre \tau ^na).
&\hphantom{(6.12)\kern7pt}{\rm (ii)}\cr}
$$

{\it Proof}\pointir
It is enough to prove (i), since (ii) then follows by taking inverses
and invoking~(6.6). From~(6.5) we have
$$
\displaylines{
\quad
\sum _{i=0}^{m+n} (-1)^i e_i(x,y\barre a)(t\barre a)^{m+n-i}
=\prod _{i=1}^n (t-x_i)\prod _{j=1}^m(t-y_j)\hfill\cr
\hfill
\eqalign{&=\sum _{j=0}^n(-1)^j e_j(x\barre a)(t\barre a)^{n-j}
\sum _{k=0}^m (-1)^k e_k(y\barre \tau ^{n-j}a)
(t\barre \tau^{n-j}a)^{m-k}\cr
&=\sum _{j,k}(-1)^{j+k}e_j(x\barre a)\,e_k(y\barre \tau ^{n-j}a)\,
(t\barre a)^{m+n-j-k}\cr}\quad\cr}
$$
by use of (6.1). Since the polynomials $(t\barre a)^r$, $r\ge 0$ are
linearly independent, we may equate coefficients to obtain
$$
e_i(x,y\barre a)
=\sum _{j+k=i} e_j(x\barre a)\,e_k(y\barre \tau^{n-j}a). 
$$
With a change of notation this relation takes the form
$$
(-1)^{k-i}e_{k-i}(x,y\barre \tau ^ka)
=\sum _j (-1)^{k-j} e_{k-j}(x\barre \tau ^ka)
\,(-1)^{j-i} e_{j-k}(y\barre \tau ^{n+j}a)
$$
which establishes (i).\cqfd

\par\medskip
\noindent
(6.13) {\it Let $\lambda $, $\mu $ be partitions. Then}
$$
s_{\lambda /\mu }(x,y\barre a)=\sum _\nu 
s_{\nu /\mu }(x\barre a)\,s_{\lambda /\nu }(y\barre \tau ^na).
$$

{\it Proof}\pointir
Let $r\ge \max(l(\lambda ),\,l(\mu ))$. By definition (6.8),
$s_{\lambda /\mu }(x,y\barre a)$ is the $r\times r$~minor of
${\bf H}(x,y\barre a)$ corresponding to the row indices
$\mu _1-1$, \dots~, $\mu _r-r$ and the column indices
$\lambda _1-1$, \dots~, $\lambda _r-r$, that is to say, it is the
element of
$\bigwedge ^r {\bf H}(x,y\barre a)$ indexed by these sets of indices.
The formula (6.13) now follows from (6.12)~(ii) and the
functoriality of exterior powers,\footnote{$^*$}{also known as the
Cauchy-Binet identity.} which together imply that
$\bigwedge ^r{\bf H}(x,y\barre a)=\bigwedge ^r{\bf H}(x\barre a)
.\bigwedge ^r{\bf H}(y\barre \tau ^n a)$.\cqfd
\medskip


By iterating (6.13) we obtain the following result. Let
$x^{(i)}$, \dots~, $x^{(n)}$ be $n$~sets of variables, where
$x^{(i)}=(x_1^{(1)},\ldots,x_{r_i}^{(i)})$, and let $\lambda $, $\mu $
be partitions. Then
$$
s_{\lambda /\mu }(x^{(i)},\ldots, x^{(n)}\barre a)
=\sum _{(\nu )} \prod _{i=1}^n
s_{\nu ^{(i)}/\nu ^{(i-1)}}
(x^{(i)}\barre \tau ^{r_1+\cdots +r_{i-1}}a)
\leqno(6.14)
$$
summed over all sequences
$(\nu )=(\nu ^{(0)},\ldots, \nu ^{(n)})$ of partitions,
such that $\mu =\nu ^{(0)}\subset \nu ^{(1)}\subset\cdots\subset
\nu ^{(n)}=\lambda $.\cqfd

\medskip
We shall apply (6.14) in the case that each $x^{(i)}$ consists of a
single variable~$x_i$ (so that $r_i=1$ for $1\le i\le n$). For a
single~$x$ we have $s_{\lambda /\mu }(x\barre a)=0$ unless
$\lambda -\mu $ is a horizontal strip, by (6.10) ; and if $\lambda
-\mu $ is a horizontal strip it follows from (6.8) that
$$
\eqalign{s_{\lambda /\mu }(x\barre a)&=\prod _{i\ge 1}
h_{\lambda _i-\mu _i} (x\barre \tau ^{\mu _i-i+1}a)\cr
&=\prod _{i\ge 1}(x\barre \tau ^{\mu _i-i+1}a)^{\lambda _i-\mu
_i}.\cr}
 $$
since $h_r(x\barre a)=s_{(r)}(x\barre a)=(x\barre a)^r$ in the 
case of a single~$x$, from the definition~(6.4). Hence

\par\medskip
\noindent
(6.15) {\it For a single $x$ we have
$$
s_{\lambda /\mu }(x\barre a)=\prod _{s\in \lambda -\mu }
(x+a_{c(s)+1})
$$
if $\lambda -\mu $ is a horizontal strip, and
$s_{\lambda /\mu }(x\barre a)=0$ otherwise.}

\noindent
(Here $c(s)$ is the {\it content} of $s$, i.e., $c(s)=j-i$ if $s=(i,j)$.)

\medskip
From (6.14) and (6.15) it now follows that if 
$x=(x_1,\ldots ,x_n)$
$$
s_{\lambda /\mu }(x\barre a)=\sum _T (x\barre a)^T
\leqno(6.16)
$$
{\it summed over column strict tableaux
$T:\lambda -\mu \rightarrow [1,n]$, where}
$$
(x\barre a)^T=\prod _{s\in \lambda -\mu } 
\bigl(x_{T(s)}+a_{T^*(s)}\bigr)
$$
{\it and} $T^*(i,j)=T(i,j)+j-i$
(so that $T^*$ is row-strict).

\medskip
When $a_i=1-i$ for all $i\in {\bboard Z}$ (Variation 4), (6.16) reduces to
the definition~(4.2) of the factorial Schur functions.

\medskip
Finally, there is an analogue of the dual Cauchy formula : namely (with
the notation of $(0.11')$)
$$
\prod _{i=1}^n \prod _{j=1}^m
(x_i+y_j)=\sum _\lambda  s_\lambda (x\barre a)
\,s_{\widehat \lambda '}(y\barre -a)\leqno(6.17)
$$
{\it where $-a$ is the sequence $(-a_n)_{n\in {\bboard Z}}$.}

\medskip
{\it Proof}\pointir
Consider the quotient
$$
A_{\delta _{m+n}}(x,y)\bigm/ A_{\delta _n}(x)A_{\delta _m}(y)
$$
which by (6.3) is equal to
$\prod\limits _{i,j}(x_i-y_j)$. On the other hand, Laplace expansion
of the determinant
$A_{\delta _{m+n}}(x,y)$ gives
$$
A_{\delta _{m+n}}(x,y)=\sum _{\lambda \subset (m^n)}
(-1)^{\left |\widehat  \lambda \right |}
A_{\lambda +\delta _n}(x)\,
A_{\widehat  \lambda '+\delta _m}(y).
$$
Hence we have
$$
\prod _{i,j}(x_i-y_j)=\sum _{\lambda \subset (m^n)}
(-1)^{\left |\widehat  \lambda \right |}
s_\lambda (x\barre a)\,s_{\widehat  \lambda '}(y\barre a)
$$
and by replacing each $y_j$ by $-y_j$ we obtain (6.17).\cqfd

\rem Remark|From the definition (6.1) it follows that
$$
(x\barre a)^r=\sum _{k\ge 0} x^k e_{r-k}\bigl(a^{(r)}\bigr), 
$$
where $a^{(r)}=(a_1,a_2,\ldots, a_r)$. Hence, with
$x=(x_1,\ldots, x_n)$,
$$
\eqalign{A_\alpha (x\barre a)&=\det \Bigl(\sum _{\beta _k\ge 0}
x_i^{\beta _k} e_{\beta _k-\alpha _j}\bigl(a^{(\alpha _j)}\bigr)\Bigr)\cr
&=\sum _\beta  \det \bigl(x_i^{\beta _k}\bigr)
\det\Bigl(e_{\beta _k-\alpha _j}\bigl(a^{(\alpha _j)}\bigr)\Bigr)\cr} 
$$
summed over $\beta =(\beta _1,\ldots, \beta _n)\in {\bboard N}^n$ such that
$\beta _1>\beta _2>\cdots >\beta _n$.

On dividing both sides by the Vandermonde determinant
 $\Delta (x)$ and replacing $\alpha $, $\beta $ by $\lambda +\delta $, $\mu +\delta $ respectively, we
obtain
$$
s_\lambda (x\barre a)=\sum _{\mu \subset \lambda } s_\mu (x)
\det\Bigl(e_{\lambda _i-\mu _j-i+j}\bigl(a^{(\lambda _j+n-j)}\bigr)\Bigr),
\leqno(6.18)
$$
symmetric in the $x$'s but not in the $a$'s.

Now assume that the $a$'s are independent variables ; then we
can let $n\rightarrow \infty$ (which would not have been
possible in the contexts of Variations~4 and~5). In the limit
the right-hand side of (6.18) becomes, by virtue of~(0.8),
$$
\sum _{\mu \subset \lambda } s_\mu (x)s_{\lambda '/\mu '}(a) 
$$
where $x=(x_1,x_2,\ldots\,)$ and $a=(a_1,a_2,\ldots\,)$. It
follows that
$$
\lim_{n\rightarrow \infty} s_\lambda (x_1,\ldots, x_n\barre a)=
s_\lambda (x\babarre a),\leqno(6.19) 
$$
where $s_\lambda (x\babarre a)$ is the ``supersymmetric Schur function"
defined by
$$
s_\lambda (x\babarre a)=\det \bigl(h_{\lambda _i-i+j}(x\babarre a)\bigr) 
$$
in which $h_r(x\babarre a)$ is the coefficient of $t^r$ in the
power series expansion of
$\prod \limits_{i\ge 1}(1-tx_i)^{-1}\prod \limits_{j\ge 1}(1+ta_j)$.
Thus the limit as $n\rightarrow \infty$ of $s_\lambda (x_1,\ldots
,x_n\barre a)$ is symmetric in the $a$'s as well as in the $x$'s.
From (6.19) and (6.16) we conclude that, with the notation of
(6.16),
$$
s_\lambda (x\babarre a)=\sum _T (x\barre a)^T\leqno (6.20) 
$$
summed over all column-strict tableaux $T$ of shape~$\lambda $ with
positive integer entries.

For the skew functions the corresponding result reads as
follows. Let $x=(x_n)_{n\in {\bboard Z}}$,
$a=(a_n)_{n\in {\bboard Z}}$ now be two doubly infinite
sequences of independent variables, and let $\lambda $, $\mu $ be
partitions such that $\lambda \supset \mu $. The ``skew supersymmetric
Schur function" $s_{\lambda /\mu }(x\babarre a)$ is defined by
$$
s_{\lambda /\mu } (x\babarre a)=\det \bigl(h_{\lambda _i-\mu _j-i+j}(x\babarre
a)\bigr),
 $$
where $h_r(x\babarre a)$ is now the coefficient of $t^r$ in the
power series expansion of 
$\prod \limits_{i\in {\bboard Z}}(1-tx_i)^{-1}
\prod \limits_{j\in{\bboard Z}}(1+ta_j)$. Then we have
$$
s_{\lambda /\mu }(x\babarre a)=\sum _T(x\barre a)^T \leqno(6.21) 
$$
{\it summed over all column-strict tableaux}
$T:\lambda -\mu \rightarrow {\bboard Z}$. (6.20) and (6.21) were found
independently by Ian Goulden and Curtis Greene.

\titre 7\thinspace th Variation|
Here we shall work over a finite field $F={\bboard F}_q$ of
cardinality~$q$ (so that $q$ is a prime power). Let $x_1$, \dots~, $x_n$
be independent indeterminates over~$F$, and let 
$V\subset F[x_1,\ldots,x_n]$ denote the $F$-vector space spanned by the
$x_i$, so that
$F[x_1,\ldots,x_n]$ is the symmetric algebra ${\bf S}(V)$ of~$V$
over~$F$.

For each $\alpha =(\alpha _1,\ldots,\alpha _n)\in {\bboard N}^n$ we define
$$\displaylines{\rlap{(7.1)}\hfill
A_\alpha =\det\bigl(x_i^{q^{\alpha _j}}\bigr)_{1\le i,j\le n}.
\hfill\cr
\noalign{\hbox{If $v\in V$, $v\not=0$, so that}}
\rlap{(7.2)}\hfill
v=a_1x_1+\cdots+a_nx_n\hfill\cr
\noalign{\hbox{with coefficients $a_i\in F$, not all zero, then we have}}
v^{q^r}=a_1x_1^{q^r}+\cdots+a_nx_n^{q^r}\cr
\noalign{\vskip -8pt}}
$$
for all integers $r\ge 0$, from which it follows that the determinant
(7.1) is divisible by~$v$ in~${\bf S}(V)$. Hence if $V_0$ is the subset
of~$V$ consisting of all the vectors~(7.2) for which the first non zero
coefficient~$a_i$ is equal to~1, we see that $A_\alpha $ is divisible
in~${\bf S}(V)$ by the product
$$
P=P(x_1,\ldots,x_n)=\prod _{v\in V_0} v,\leqno(7.3)
$$
which is homogeneous of degree
$$
{\rm Card}(V_0)=q^{n-1}+q^{n-2}+\cdots+1.
$$

In particular, when $\alpha =\delta _n=\delta =(n-1,n-2,\ldots, 1,0)$,
$A_\delta $ is divisible by~$P$, and is a homogeneous polynomial of the
same degree
$q^{n-1}+q^{n-2}+\cdots+1$ ; moreover the leading term in each of~$P$
and~$A_\delta $ is the monomial
$x_1^{q^{n-1}}x_2^{q^{n-2}}\ldots x_n$, and therefore
$$
P=A_\delta .\leqno(7.4)
$$

The determinant $A_\alpha $ clearly vanishes if any two of the $\alpha
_i$ are equal, and hence (up to sign) we may assume that
$\alpha _1>\cdots>\alpha _n\ge 0$, i.e., that
$\alpha =\lambda +\delta $ where 
$\lambda =(\lambda _1, \ldots,\lambda _n)$ is a partition of
${\rm length}\le n$. It follows from what we have just proved that
$$
S_\lambda (x_1,\ldots, x_n)=A_{\lambda +\delta }\bigm/ A_\delta 
\leqno(7.5)
$$
is a polynomial, i.e., an element of ${\bf S}(V)$, homogeneous of degree
$$
\sum _{i=1}^n (q^{\lambda _i}-1) q^{n-i}.
$$

These polynomials $S_\lambda $ (and their skew analogues $S_{\lambda
/\mu }$ that we shall define later) constitute our $7\,$th  Variation.
Clearly they are symmetric in $x_1$, \dots~, $x_n$ ; but they are in fact
invariant under a larger group, namely the group
$GL_n(F)$ (or~$GL(V)$).

For if $g=(g_{ij})\in GL_n(F)$, we have
$$\eqalignno{
gx_i&=\sum _{k=1}^n g_{ki}x_k\cr
\noalign{\hbox{and therefore}}
(gx_i)^{q^r}&=\sum _k  g_{ki}x_k^{q^r}\cr
}
$$
for all integers $r\ge 0$, from which it follows that
$gA_\alpha =(\det g)A_\alpha $ and hence that
$$
S_\lambda (gx_1,\ldots,gx_n)=S_\lambda (x_1,\ldots,x_n).
$$
Consequently $S_\lambda (x_1,\ldots,x_n)$ depends only on
($\lambda $ and) the vector space~$V$, and not on the particular basis
$x_1$, \dots~, $x_n$ of~$V$, and accordingly we shall write
$S_\lambda (V)$ in place of
$S_\lambda (x_1,\ldots,x_n)$ from now on. 

When $\lambda =(r)$ we shall write
$$
H_r(V)=S_{(r)}(V)\qquad (r\ge 0)
$$
with the usual convention that
$H_r(V)=0$ if $r<0$ ; and when 
$\lambda =(1^r)$ $(0\le r\le n)$ we shall write
$$
E_r(V)=S_{(1^r)}(V)\qquad (0\le r\le  n)
$$
with the convention that $E_r(V)=0$ if $r<0$ or $r>n$.

A well-known theorem of Dickson states that the subalgebra of
$GL(V)$-invariant elements of ${\bf S}(V)$ 
is a polynomial algebra over~$F$,
generated by the
$E_r(V)$ $(1\le r\le n)$. But by contrast with the classical situation, the
$S_\lambda (V)$ do not form an $F$-basis of
$S(V)^{GL(V)}$, as one sees already in the simplest case~$n=1$.

Let $t$ be another indeterminate and let
$$
f_V(t)=\prod _{v\in V} (t+v).\leqno(7.6)
$$
From (7.3) and (7.4) it follows that
$$
\eqalign{f_V(t)&=P(t,x_1,\ldots,x_n)/P(x_1,\ldots,x_n)\cr
&=A_{\delta _{n+1}}(t,x_1,\ldots,x_n)/ A_{\delta _n}(x_1,\ldots,x_n).\cr}
$$
By expanding the determinant $A_{\delta _{n+1}}$ along the top row, we
shall obtain
$$
f_V(t)=t^{q^n}-E_1(V)t^{q^{n-1}}+\cdots +(-1)^n E_n(V)t.\leqno(7.7)
$$
Since $(at+bu)^{q^r}=at^{q^r}+bu^{q^r}$ for all $a,b\in F$ and
integers~$r\ge 0$ ($t,u$ being indeterminates) it follows from (7.7) that
$$
f_V(at+bu)=af_V(t)+bf_V(u),\leqno(7.8)
$$
i.e., that $f_V$ is an {\it additive} (or Ore) polynomial.

\bigskip
\goodbreak
Let $\varphi :{\bf S}(V)\rightarrow {\bf S}(V)$ denote the Frobenius map,
namely
$$
\varphi (u)=u^q\qquad (u\in {\bf S}(V)).
$$
The mapping $\varphi $ is an $F$-algebra endomorphism of ${\bf S}
(V)$, its image being $F[x_1^q,\ldots,x_n^q]$. Since we shall later
encounter negative powers of~$\varphi $, it is convenient to introduce
$$
\widehat {\bf S}(V)=\bigcup_{r\ge 0} {\bf S}(V)^{q^{-r}}
$$
where ${\bf S}(V)^{q^{-r}}=F[x_1^{q^{-r}},\ldots,x_n^{q^{-r}}]$.
On $\widehat  {\bf S}(V)$, $\varphi $ is an automorphism.

\bigskip
Let ${\bf E}(V)$, ${\bf H}(V)$ be the (infinite) matrices
$$
\eqalign{
{\bf H}(V)&=\bigl(\varphi ^{i+1} H_{j-i}(V)\bigr)_{i,j\in {\bboard Z}},\cr
{\bf E}(V)&
=\bigl((-1)^{j-i}\varphi ^{j} E_{j-i}(V)\bigr)_{i,j\in {\bboard Z}}.\cr} 
$$
Both are upper triangular, with 1's on the diagonal. They are related by
$$
{\bf E}(V)={\bf H}(V)^{-1}.\leqno(7.9)
$$

{\it Proof}\pointir
We have to show that
$$
\sum _j(-1)^{k-j} \varphi ^k(E_{k-j})\varphi ^{i+1}(H_{j-i})=\delta _{ik}
$$
for all $i,k$. This is clear if $i\ge k$. If $i<k$, we may argue as follows :
since $f_V(x_i)=0$ it follows from (7.7) that
$$\displaylines{
\varphi ^n(x_i)-E_1\varphi ^{n-1}(x_i)+\cdots+(-1)^nE_nx_i=0\cr
\noalign{\hbox{and hence that}}
(1)\qquad
\varphi ^{n+r-1}(x_i)-\varphi ^{r-1}(E_1)\varphi ^{n+r-2}(x_i)\hfill\cr
\hfill{}+\cdots+
(-1)^n \varphi ^{r-1}(E_n) \varphi ^{r-1}(x_i)=0\qquad\cr}
$$
for all $r\ge 0$ and $1\le i\le n$. On the other hand, by expanding the
determinant $A_{(r)+\delta }$ down the first column, it is clear that
$H_r=H_r(V)$ is of the form
$$
H_r=\sum _{i=1}^n u_i \varphi ^{n+r-1}(x_i)\leqno(2)
$$
with coefficients $u_i\in F(x_1,\ldots,x_n)$ independent of~$r$. From (1)
and (2) it follows that
$$
H_r-\varphi ^{r-1}(E_1)H_{r-1}+\cdots+
(-1)^n\varphi ^{r-1}(E_n)H_{r-n}=0
\leqno(3)
$$
for each $r\ge 0$. Putting $r=k-i$ and operating on (3) with 
$\varphi ^{i+1}$, we obtain
$$
\sum _{i\le j\le k} (-1)^{k-j} \varphi ^k(E_{k-j})
\varphi ^{i+1}(H_{j-i})=0
$$
as required.\cqfd

Next, we have analogues of the Jacobi-Trudi and N\"agelsbach-Kostka
formulas (0.2), (0.3) :

\par
\medskip
\noindent
(7.10) {\it Let $\lambda $ be a partition of ${\it length}\le n=\dim V$.
Then   }
$$
\eqalign{S_\lambda (V)&=\det\bigl(\varphi ^{1-j}H_{\lambda
_i-i+j}(V)\bigr)\cr
&=\det\bigl(\varphi ^{j-1}E_{\lambda '_i-i+j}(V)\bigr).\cr}
 $$

\medskip
{\it Proof}\pointir
Let $\alpha =(\alpha _1,\ldots ,\alpha _n)\in {\bboard N}^n$. From
equation~(2) above we have
$$
\varphi ^{1-j}(H_{\alpha _i-n+j})=\sum _{k=1}^n \varphi ^{\alpha _i}(x_k)
\varphi ^{1-j}(u_k)\qquad (1\le i,j\le n)
$$ 
 which shows that the matrix 
$\bigl(\varphi ^{1-j}H_{\alpha _i-n+j}\bigr)_{i,j}$
is the product of the matrices
$\bigl(\varphi ^{\alpha _i}x_k\bigr)_{i,k}$
and
$\bigl(\varphi ^{1-j}u_k\bigr)_{k,j}$.
On taking determinants it follows that
$$
\det\bigl(\varphi ^{1-j}H_{\alpha _i-n+j}\bigr)=A_\alpha B\leqno(1)
$$
where $B=\det\bigl(\varphi ^{1-j}u_k\bigr)$.

In particular, taking $\alpha =\delta $ (so that $\alpha _i-n+j=j-i$), the
left-hand side of (1) becomes equal to~1, so that $A_\delta B=1$ and
therefore
$$
\det\bigl(\varphi ^{1-j} H_{\alpha _i-n+j}\bigr)
=A_\alpha \,/\,A_\delta 
$$
for all $\alpha \in {\bboard N}^n$. Taking $\alpha =\lambda +\delta $, we
obtain the first of the formulas (7.10). The second formula (involving
the~$E$'s) is then deduced from it and (7.9), exactly as in the case of
Schur functions ($[{\rm M}_1]$, Ch.~I \S\kern 2pt 2).\cqfd

\medskip
More generally, if $\lambda $ and $\mu $ are partitions we define
$$
S_{\lambda /\mu }(V)
=\det\bigl(\varphi ^{\mu _j-j+1} H_{\lambda _i-\mu_j-i+j}(V)\bigr) 
\leqno(7.11)
$$
and then it follows as above from (7.9) that also
$$
S_{\lambda /\mu }(V)
=\det\bigl(\varphi ^{-\mu' _j+j-1} E_{\lambda' _i-\mu'_j-i+j}(V)\bigr).
\leqno(7.11')
$$
Moreover
$$
S_{\lambda /\mu }=0\ {\it unless}\ 0\le \lambda _i'-\mu '_i\le n\ {\it
for\ all\ }i\ge 1.
\leqno(7.12)
$$

The proof is the same as for Schur functions
($[{\rm M}_1]$, Ch.~I \S\kern 2pt 5).

\medskip
The hook and ribbon formulas (0.4), (0.5) remain valid in the present
context : if 
$\lambda =(\alpha _1,\ldots,\alpha _p\barre \beta _1,\ldots, \beta _p)$
in Frobenius notation, then
$$
\eqalign{S_\lambda (V)&=\det \bigl(S_{(\alpha _i\barre \beta _j)}
(V)\bigr)_{1\le i,j\le p}\cr
&=\det \bigl(S_{[\alpha _i\barre \beta _j]}
(V)\bigr)_{1\le i,j\le p}\cr
}
$$
in the notation of (0.5). These identities are formal consequences of the
definition (7.11) :
see \S\kern 2pt 9 below.

\rem Remark| Since $H_r(V)$ has degree $(q^r-1)q^{n-1}$, it follows that
the degree of
$\varphi ^{\mu _j-j+1}H_{\lambda _i-\mu _j-i+j}(V)$ is
$$
q ^{\mu _j-j+1}(q^{\lambda _i-\mu _j-i+j}-1)q^{n-1}=q^{\lambda
_i+n-i}-q^{\mu _j+n-j}.
 $$
Hence the determinant (7.11) (and likewise (7.12)) is isobaric, and
$S_{\lambda /\mu }(V)$ is homogeneous of degree
$$
\sum _{i=1}^n (q^{\lambda _i}-q^{\mu _i})q^{n-i}.\leqno(7.13)
$$

We shall next consider analogues of the ``addition formula"
$$
s_{\lambda /\mu }(x,y)=\sum _\nu  s_{\lambda /\mu }(x)s_{\nu /\mu }(y)
$$
for Schur functions 
($[{\rm M}_1]$, Ch.~I \S\kern 2pt 5).
For this purpose let $U$ be an $F$-vector subspace of~$V$. From (7.8) it
follows that
$$
f_U:v\mapsto f_U(v)=\prod _{u\in U}(v+u)
$$
is an $F$-linear mapping of $V$ into ${\bf S}(V)$ with kernel~$U$. Hence
$f_U(V)$ is isomorphic to the quotient of~$V$ by~$U$, and we shall write
$$
f_U(V)=V/U\leqno(7.14)
$$
thereby reserving the notation $V/U$ for this particular embedding of the
quotient space in~${\bf S}(V)$. If $U'$ is a vector space complement
of~$U$ in~$V$, the elements of~$V/U$ are the products 
$f_U(v)=\prod\limits _{u\in U}(v+u)$ as $v$ runs through~$U'$, that is to
say, they are the products in~${\bf S}(V)$ of the elements of the cosets
of~$U$ in~$V$.

With this notation we have
$$
f_V=f_{V/U}\circ f_U\qquad
\bigl[{\it i.e.,}\ f_V(t)=f_{V/U}(f_U(t))\bigr].
\leqno(7.15)
$$

{\it Proof}\pointir
We have
$$\eqalignno{
f_{V/U}\bigl(f_U(t)\bigr)&=\prod _{w\in V/U}\bigl(f_U(t)+w\bigr)\cr
&=\prod _{u'\in U'} \bigl(f_U(t)+f_U(u')\bigr)\cr
&=\prod _{u'\in U'}f_U(t+u')&[{\rm by}\ (7.8)]\cr
&=\prod _{\scriptstyle u\in U\atop \scriptstyle u'\in U'}
(t+u+u')=f_V(t).\qed\cr}
$$


\medskip
\noindent
(7.16) {\it Let $T$ be a vector subspace of $U$. Then}
$$
V/U=(V/T)\bigm/(U/T)
$$
(with equality, not merely isomorphism).

\medskip
{\it Proof}\pointir
By definition
$$
\eqalignno{
(V/T)/(U/T)&=f_{U/T}(V/T)=f_{U/T}(f_T(V))\cr
&=f_U(V)=V/U.\qed&[{\rm by\ (7.15)}]\cr
}
$$

$$\displaylines{
(7.17)\quad  {\rm (i)}\quad 
{\bf E}(V)=\varphi ^{\dim(V/U)}\bigl({\bf E}(U)\bigr)\cdot {\bf
E}(V/U),\hfill\cr 
\hphantom{(7.17)\kern7pt}  {\rm (ii)}\quad 
{\bf H}(V)={\bf H}(V/U)\cdot\varphi ^{\dim(V/U)} 
\bigl({\bf H}(U)\bigr).\hfill\cr}
$$

{\it Proof}\pointir
It is enough to prove (i), since (ii) then follows by taking inverses and
using~(7.9). From (7.7) and (7.15) we have ($\dim V=n$, $\dim U=m$)

\goodbreak

$$
\displaylines{
\quad
\sum _{i\ge 0} (-1)^i E_i(V)
\varphi ^{n-i}(t)=f_V(t)=f_{V/U}\bigl(f_U(t)\bigr)\hfill\cr
\hfill\eqalign{&=\sum _{j\ge 0} (-1)^j E_j(V/U) \varphi ^{n-m-j}
\Bigl(\sum _{k\ge 0} (-1)^k E_k(U) \varphi ^{m-k}(t)\Bigr)\cr
&=\sum _{j,k\ge 0}(-1)^{j+k} E_j(V/U)\varphi ^{n-m-j} \bigl(E_k(U)\bigr)
\varphi ^{n-j-k}(t)\cr} \quad\cr
\noalign{\hbox{and therefore}}
E_i(V)=\sum _{j+k=i} E_j(V/U) \varphi ^{n-m-j}\bigl(E_k(U)\bigr).\cr
\noalign{\hbox{With a change of notation this can be written in the form}}
\noalign{\vskip 5pt}
E_{c-a}(V)=\sum _{a\le b\le c}\varphi ^{n-m+b-c}\bigl(E_{b-a}(U)\bigr)
E_{c-b}(V/U)\cr
\noalign{\hbox{or equivalently}}
\noalign{\vskip 5pt}
\quad (-1)^{c-a}\varphi ^c\bigl(E_{c-a}(V)\bigr)\hfill\cr
\hfill{}=\sum _b(-1)^{b-a} \varphi ^{n-m+b}\bigl(E_{b-a}(U)\bigr)
(-1)^{c-b}\varphi ^c\bigl(E_{c-b}(V/U)\bigr),\quad\cr
 }
$$
proving (i).\cqfd

\par\bigskip
\noindent
(7.18) {\it Let $\lambda ,\mu $ be partitions. Then}
$$
S_{\lambda /\mu }(V)=\sum _\nu S_{\nu /\mu }(V/U)\cdot
\varphi ^{\dim(V/U)}\bigl(S_{\lambda /\mu }(U)\bigr).
$$

{\it Proof}\pointir
Suppose
$r\ge \max\bigl(l(\lambda ), l(\mu )\bigr)$. By definition (7.11), 
$S_{\lambda /\mu }(V)$ is the $r\times r$ minor of~${\bf H}(V)$
corresponding to the row indices $\mu _1-1$, \dots~, $\mu _r-r$ and
the column indices 
$\lambda _1-1$, \dots~, $\lambda _r-r$, that is to say, it is the element
of
$\bigwedge\limits^r{\bf H}(V)$ indexed by these sets of indices.
The formula (7.18) now follows from (7.17) (ii) and the functoriality of
exterior powers, which together imply that
$\bigwedge\limits^r{\bf H}(V)=\bigwedge\limits^r{\bf H}(V/U)\cdot
\varphi ^{\dim(V/U)}\bigwedge\limits^r{\bf H}(U)$.\cqfd

\bigskip
By iteration of (7.18) (and making use of (7.16)) we obtain

\par\medskip
\noindent
(7.19) {\it Let  $V_0>V_1>\cdots>V_r$ be a chain of subspaces of $V$.
Then}
$$
S_{\lambda /\mu }(V_0/V_r)
=\sum _{(\nu )} \prod _{i=1}^r
\varphi ^{\dim(V_0/V_{i-1})}
S_{\nu ^{(i)}/\nu ^{(i-1)}}(V_{i-1}/V_i)
$$
{\it summed over all sequences 
$(\nu ) =(\nu ^{(0)},\nu ^{(1)}, \ldots, \nu ^{(r)})$ of partitions such that}
$\mu =\nu ^{(0)}\subset \nu ^{(1)}\subset\cdots\subset\nu ^{(r)}
=\lambda $ .\cqfd

\bigskip
As in the case of Schur functions, we shall use (7.19) to express
$S_{\lambda /\mu }(V)$ as a sum over column-strict tableaux of shape
$\lambda -\mu $.

If $U$ is any finite-dimensional subspace of $F(V)$, let
$$
\pi (U)=\prod_ {\scriptstyle u\in U\atop \scriptstyle u\not=0} u.
$$
From (7.7) it follows that
$$
\pi (U)=(-1)^d E_d(U)\leqno(7.20)
$$
if $\dim U=d$.

If $U'$ is a subspace of $U$, then $U/U'$ is defined by (7.14) as a subspace
of
${\bf S}(U)\subset {\bf S}(V)$, and its elements are the products
(in ${\bf S}(U)$ or ${\bf S}(V)$) of the elements of the cosets of~$U'$
in~$U$. From this it follows that
$$
\pi (U/U')=\pi (U)/\pi (U')=\prod _{u\in U-U'}u.\leqno(7.21)
$$

We now consider the case where $U$ is 1-dimensional.

\par\medskip
\noindent
(7.22) {\it Let $U$ be a $1$-dimensional subspace of ${\bf S}(V)$ and let
$\lambda ,\mu $ be partitions. Then}
$$
S_{\lambda /\mu }(U)=(-1)^{\left |\lambda -\mu \right |}
\prod _{s\in \lambda -\mu } \varphi ^{c(s)}\pi (U)
$$
{\it if $\lambda -\mu $ is a horizontal strip, and is zero otherwise.}

\noindent
(Here $c(s)$ is the {\it content} of $s$ :
$c(s)=j-i$ if $s=(i,j)$.)

\medskip
{\it Proof}\pointir
Since $\dim U=1$ we have $E_r(U)=0$ for $r\ge 2$ and hence, by (7.10),
$$
\eqalign{H_r(U)=S_{(r)}(U)&=\prod _{j=1}^r \varphi ^{j-1}E_1(U)\cr
&=(-1)^r \prod _{j=1}^r \varphi ^{j-1}\pi (U)\cr
}
$$
by (7.20). From (7.12), we have $S_{\lambda /\mu }(U)=0$ unless
$0\le \lambda _i'-\mu '_i\le 1$ for all~$i$, that is to say, unless
$\lambda -\mu $ is a horizontal strip. If on the other hand
$\lambda -\mu $ is a horizontal strip, we have
$$
\eqalign{
S_{\lambda /\mu }(U)&=\det\bigl(\varphi ^{\mu _j-j+1}
H_{\lambda _i-\mu _j-i+j}(U)\bigr)\cr
&=\prod _{i\ge 1}\varphi ^{\mu _i-i+1} H_{\lambda _i-\mu _i}(U)\cr
&=(-1)^{\left |\lambda -\mu \right |}
\prod _{i\ge 1} \varphi ^{\mu _i-i+1}
\prod _{j=1}^{\lambda _i-\mu _i} \varphi ^{j-1}\pi (U)\cr
&=(-1)^{\left |\lambda -\mu \right |}
\prod _{s\in \lambda -\mu } \varphi ^{c(s)} \pi (U).\qed\cr
}
$$

\def\gov{{\goth V}}

Now let
$$
{\goth V}:\quad V=V_0>V_1>\cdots>V_n=0
$$
be a (full) flag in $V$, so that $V_i$ is a vector subspace of~$V$ of 
dimension~$(n-i)$, for each~$i$. Let
$$
\pi _i({\goth V})=\pi (V_{i-1}/V_i)=\prod _{v\in V_{i-1}-V_i} v
$$ 
for $1\le i\le n$.

From (7.19) and (7.22) it follows that
$$
\eqalign{
S_{\lambda /\mu }&=\sum _{(\nu )} \prod _{i=1}^n
\varphi ^{i-1}S_{\nu ^{(i)}/\nu ^{(i-1)}}(V_{i-1}/V_i)\cr
&=(-1)^{|\lambda -\mu |} \sum _{(\nu )}
\prod _{i=1}^n \prod _{s\in \nu ^{(i)}/\nu ^{(i-1)}}
\varphi ^{i-1+c(s)}\pi _i({\goth V})\cr
}
$$
summed over all sequences $(\nu )=(\nu ^{(0)},
\nu ^{(1)},\ldots,\nu ^{(n)})$ of partitions such that
$\mu  =\nu ^{(0)}\subset
\nu ^{(1)}\subset\cdots\subset \nu ^{(n)}=\lambda $ and each
$\nu ^{(i)}-\nu ^{(i-1)}$ is a horizontal strip. Such sequences are in
one-one correspondence with column-strict tableaux
$T:\lambda -\mu \rightarrow [1,n]$, and hence we obtain
$$
S_{\lambda /\mu }(V)=(-1)^{| \lambda -\mu |}\sum _T \psi(T,\gov)
\leqno(7.23)
$$
{\it summed over column strict tableaux
$T:\lambda -\mu \rightarrow [1,n]$, where
$$
\psi (T,\gov)=\prod _{s\in \lambda -\mu } \varphi ^{T^*(s)-1}
\pi_{T(s)}(\gov),$$
and} $T^*(i,j)=T(i,j)+j-i$ (as in (6.16)).

\goodbreak
\rem Remark|The degree of $\psi (T,\gov)$ is
$$
 \sum _{s\in \lambda -\mu } q^{c(s)+T(s)-1}
(q^{n-T(s)+1}-q^{n-T(s)})
=\sum _{s\in \lambda -\mu }
q^{c(s)}(q^n-q^{n-1})
$$
which is easily seen to be equal to
$\smash{\sum \limits_{i=1}^n(q^{\lambda _i}-q^{\mu _i})q^{n-i}}$, in
agreement with (7.13).

\bigskip
In the formula (7.23) the flag $\gov$ is fixed and the sum is over
tableaux~$T$. Since $S_{\lambda /\mu }(V)$ is
$GL(V)$-invariant, and since the number of flags $\gov$ in~$V$ is
congruent to~1 modulo~$q$, we may sum over all flags as well :
$$
S_{\lambda /\mu }(V)=(-1)^{ |\lambda -\mu |} \sum _{T,\gov}
\psi (T,\gov),\leqno(7.23')
$$

It seems plausible that (when $\mu =0$) there should be another
expression for $S_\lambda (V)$ as a sum over flags, namely
$$
S_\lambda (V)=(-1)^{| \lambda |}\sum _{\gov} \psi (T_0,\gov)
\leqno(7.24\,{\rm ?})
$$
where $T_0$ is the tableau defined by
$T_0(i,j)=i$ for all $(i,j)\in \lambda $. For this tableau we have
$$
\eqalign{
\psi (T_0,\gov)&=\prod _{(i,j)\in \lambda } 
\psi ^{j-1} \pi_i(\gov)\cr 
&=(-1)^{|\lambda |}\prod _{i\ge 1} H_{\lambda _i}(V_{i-1}/V_i).\cr}
$$
The formula (7.24 ?) is true for example when
$\lambda =(1^r)$ $(1\le r\le n)$, and in some other cases ; but I do
not know whether it is true generally. It would be enough to show
that, if $l(\lambda )<n$,
$$
S_\lambda (V)=\sum _L S_\lambda (V/L)\leqno(7.25\,{\rm ?})
$$
{\it summed over all lines} ({\it i.e., $1$-dimensional
subspaces})~$L$ {\it in}~$V$.

\bigskip
Finally, we shall indicate an analogue of the dual Cauchy formula
$(0.11')$. Let $V$ (resp. $W$) be the $F$-vector space spanned by
$x_1$, \dots~,~$x_n$ (resp. $y_1$, \dots~, $y_m$), the $x$'s and
$y$'s being independent indeterminates over~$F$. Let
$$
\pi (V,W)=\prod _{L,M} \bigl( \pi (L)+\pi (M)\bigr)
$$
where the product on the right is over all lines~$L$ in~$V$ and~$M$
in~$W$. Then we have (with the notation of $(0.11')$)
$$
\pi (V,W)=\sum _\lambda S_\lambda (V) S_{\widehat \lambda '}(W).
\leqno(7.26)
$$

{\it Proof}\pointir
Consider the quotient
$$
A_{\delta _{m+n}}(x,y)\bigm/ A_{\delta _n}(x)A_{\delta _m}(y)
$$
which by (7.4) is equal to the product
$$
\eqalignno{
\prod _{{\scriptstyle v\in V_0\atop \scriptstyle w\in W}\atop
\scriptstyle w\not=0}(v+w)
&=\prod _{\scriptstyle w\in V_0\atop\scriptstyle w\in W_0}
(v^{q-1}-w^{q-1})\cr
&=\prod _{L,M}\bigl(\pi (L)-\pi (M)\bigr)&(1)\cr
}
$$
(product over lines $L<V$ and $M<W$). On the other hand, by Laplace
expansion of the determinant
$A_{\delta _{m+n}}(x,y)$, we obtain
$$
A_{\delta _{m+n}}(x,y)=\sum _{\lambda \subset (m^n)}
(-1)^{|\widehat \lambda |} A_{\lambda +\delta _n}(x)
A_{\widehat \lambda '+\delta _m}(y).\eqno(2)
$$
From (1) and (2) it follows that
$$
\prod _{L,M}\bigl(\pi (L)-\pi (M)\bigr)
=\sum _{\lambda \subset (m^n)} 
(-1)^{|\widehat \lambda|}
S_\lambda (V) S_{\widehat \lambda '}(W).
$$
Finally, to get rid of the minus signs, replace each $y_j$ by 
$\omega y_j$, where $\omega $ lies in an extension field of~$F$ and
satisfies
$\omega ^{q-1}=-1$.\cqfd

\vskip30pt minus 5pt
\centerline{\bf 8\thinspace th Variation : flagged Schur functions}
\vskip5pt

Let $x_1$, $x_2$, \dots\ be independent variables. For all positive
integers
$a$, $b$, $r$ define $h_r(a,b)$ (resp.   $e_r(a,b)$) to be the complete
(resp. elementary) symmetric function of degree~$r$ in the
variables $x_a$, $x_{a+1}$, \dots~, $x_b$ if $a\le b$, and to be zero
if $a>b$ ; also define
$h_0(a,b)=e_0(a,b)=1$ for all $a,b$, and $h_r(a,b)=e_r(a,b)=0$ for
all~$a$, $b$ when $r<0$.

Let $\lambda $, $\mu $ be partitions of ${\rm length}\le n$ and let
$a=(a_1,\ldots,a_n)$, $b=(b_1,\ldots,b_n)$ be sequences of positive
integers. The {\it row-flagged Schur function} 
$s_{\lambda /\mu}(a,b)$ with row flags $a$, $b$ is defined [W] to~be
$$
s_{\lambda /\mu }(a,b)=\det\bigl(h_{\lambda _i-\mu
_j-i+j}(a_j,b_i)\bigr)_{1\le i,j\le n}\leqno(8.1)
 $$
It is zero unless $\lambda \supset \mu $, which we assume
henceforth.

\goodbreak
Let $m\ge \lambda _1$ (so that $\mu \subset \lambda \subset
(m^n))$ and let $a'=(a'_1,\ldots, a_m')$,
$b'=(b'_1,\ldots, b'_m)$ be sequences of positive integers. The notion
dual to (8.1) is that of the {\it column-flagged Schur function}
$s'_{\lambda /\mu }(a',b')$\footnote{$^*$}{Our notation here differs
from that of [W].} with column flags $(a',b')$ :
$$
s'_{\lambda /\mu }(a',b')=\det\bigl(e_{\lambda '_i-\mu '_j-i+j}
(a'_j,b'_i)\bigr)_{1\le i,j\le m}\leqno(8.1')
$$
where $\lambda '$, $\mu '$ are the partitions conjugate to 
$\lambda $, $\mu $ respectively.

With mild restrictions on the flags these Schur functions can be
expressed as sums over tableaux :

\par\medskip
\noindent
(8.2) {\it Suppose that $a_i\le a_{i+1}$ and $b_i\le b_{i+1}$
whenever
$\mu _i<\lambda _{i+1}$} (i.e., the sequences $a$, $b$ are increasing
on each connected component of $\lambda -\mu $).
{\it Then
$$
s_{\lambda /\mu }(a,b)=\sum _T x^T
$$
summed over column-strict tableaux $T$ of shape $\lambda -\mu $
such that $a_i\le T(i,j)\le b_i$ for all $(i,j)\in \lambda -\mu $,
where as usual}
$x^T=\prod \limits_{s\in \lambda -\mu }x_{T(s)}$.

\noindent
$(8.2')$ {\it Suppose that $a'_i-\mu _i'\le a'_{i+1}-\mu '_{i+1}+1$
and $b'_i-\lambda '_i\le b'_{i+1}-\lambda '_{i+1}+1$ whenever
$\mu '_i<\lambda'  _{i+1}$. Then
$$
s'_{\lambda /\mu }(a',b')=\sum _T x^T
$$
summed over column-strict tableaux $T$ of shape $\lambda -\mu $
such that
$a'_j\le T(i,j)\le b'_j$ for all $(i,j)\in \lambda -\mu $.}

\medskip
Both these results are proved in [W].

\goodbreak
\medskip
In general (i.e., for arbitrary choices of the flags $a$, $b$) the row
flagged Schur function
$s_{\lambda /\mu }(a,b)$ will not be equal to any column-flagged
$s'_{\lambda /\mu }(a',b')$. However, there is the following duality
theorem :

\par\medskip
\noindent
(8.3) {\it Let $\lambda $, $\mu $ be partitions such that
$\mu \subset \lambda \subset (m^n)$ and  let $\alpha $, $\beta $ be
positive integers such that $\alpha >m$ and $\beta -\alpha \ge m$.
Let}
$$
\displaylines{
a _i=\alpha +i-\mu _i-1,\qquad 
b_i=\beta +i-\lambda _i\qquad
(1\le i\le n),\cr
a'_j=\alpha +\mu '_j-j+1,\qquad
b'_j=\beta +\lambda '_j-j\qquad(1\le j\le m).\cr
}
$$
{\it Then} $s_{\lambda /\mu }(a,b)=s'_{\lambda /\mu }(a',b')$.

\medskip
This is a particular case of $(9.6')$ in the next section ;
alternatively, it is not hard to verify that with these choices of $a$,
$b$, $a'$, $b'$ a column-strict tableau~$T$ of shape $\lambda -\mu
$ satisfies the row restrictions of (8.2) if and only if it satisfies
the column restrictions of~$(8.2')$.

\titre 9\thinspace th Variation|
In this, our last variation, let $h_{rs}$ $(r\ge 1,\,s\in {\bboard Z})$ 
be independent indeterminates over~$\bboard Z$. Also, for convenience,
define $h_{0s}=1$ and $h_{rs}=0$ for $r<0$ and all $s\in {\bboard Z}$.
Define an automorphism of the ring~$R$ generated by the $h_{rs}$ by
$\varphi (h_{rs})=h_{r,s+1}$ for all $r,\,s$. Thus $h_{rs}=\varphi ^sh_r$,
where $h_r=h_{r0}$, and we shall generally use this alternative notation.

Now define, for any two partitions $\lambda $, $\mu $  of ${\rm length}\le
n$,
$$\leqalignno{
s_{\lambda /\mu }&=\det\bigl(\varphi ^{\mu _j-j+1} 
h_{\lambda _i-\mu _j-i+j}\bigr)_{1\le i,j\le n}&(9.1)\cr
\noalign{\hbox{and in particular $(\mu =0)$}}
s_{\lambda}&=\det\bigl(\varphi ^{-j+1} 
h_{\lambda _i-i+j}\bigr)_{1\le i,j\le n}&(9.1')\cr
}
$$

As in the case of ordinary Schur functions
($[{\rm M}_1]$, Chap.~I, \S\kern2pt 5) we have
$$
s_{\lambda /\mu }=0\qquad{\rm unless}\ \lambda \supset \mu .
\leqno(9.2)
$$

The ``Schur functions" defined by (9.1) include as special cases Variations
4, 5, 6, 7 and 8 (in part). Namely for Variation~4 we specialize
$h_{rs}=w _r(z-s)$ ; for Variation~5, 
$h_{rs}=w _r(\alpha ;z-s)$ ; for Variation~6,
$h_{rs}=h_r(x\barre \tau ^sa)$ ; for Variation~7,
$h_{rs}=\varphi ^sH_r(V)$ ; and for Variation~8,
$h_{rs}=h_r(\alpha +r+s-1,\beta +s)$.

From $(9.1')$ it follows that
$$
\displaylines{
h_r=s_{(r)}\qquad (r\ge 0)\cr
\noalign{\hbox{and we define}}
e_r=s_{(1^r)}\cr
}
$$
for all $r\ge 0$, and $e_r=0$ for $r<0$.

\goodbreak
\bigskip
\noindent
(9.3) {\it Let $I$ be any interval in ${\bboard Z}$. Then the matrices}
$$
\eqalign{
H&=H_I=\bigl(\varphi ^{1-j}h_{j-i}\bigr)_{i,j\in I}\cr
E&=E_I=\bigl((-1)^{j-i}\varphi ^{-i}e_{j-i}\bigr)_{i,j\in I}\cr
}
$$
{\it are inverses of each other.}

\medskip
{\it Proof}\pointir
Both $H$ and $E$ are upper unitriangular, hence so also are
$HE$ and $EH$. Hence it is enough to show that the $(i,k)$ element of $HE$ is
zero whenever $i,k\in I$ and $i< k$, i.e., that
$$\displaylines{
\sum _j \varphi ^{1-j}(h_{j-i})(-1)^{k-j}\varphi ^{-j}(e_{k-j})=0\cr
\noalign{\hbox{or equivalently that}}
\sum _j \varphi ^{1+i-j}(h_{j-i})(-1)^{k-j}
\varphi ^{i-j}(e_{k-j})=0.\cr
}$$

\goodbreak
\noindent
If we put $k-i=r>0$, this is equivalent to
$$\displaylines{
\rlap{(1)}\hfill
\sum _{j=0}^r (-1)^j \varphi ^{1-j}(h_j) 
\varphi ^{-j}(e_{r-j})=0.\hfill\cr
\noalign{\hbox{Now by definition}}
e_r=\det\bigl(\varphi ^{1-j}(h_{1-i+j})\bigr)_{1\le i,j\le r}\cr}
$$
and expansion of this determinant along the top row gives (1),
as re\-quired.\cqfd

\medskip
Let $\lambda $, $\mu $ be partitions such that $\lambda \supset \mu $, and
let $\theta =\lambda -\mu $. The function 
$s_\theta =s_{\lambda /\mu }$ depends not only on the skew 
shape~$\theta $ but also on its location in the lattice plane. For each 
$(p,q)\in {\bboard Z}^2$ let $\tau _{p,q}$ denote the translation
$(i,j)\mapsto (i+p,j+q)$. Then it follows immediately from (9.1) that
$$\displaylines{
s_{\tau _{0,1}(\theta )}=\varphi s_\theta ,\qquad
s_{\tau _{1,0}(\theta )}=\varphi ^{-1}s_\theta 
\cr
\noalign{\hbox{and hence that}}
\rlap{(9.4)}\hfill
s_{\tau _{p,q}(\theta )}=\varphi ^{q-p}s_\theta .\hfill\cr
}$$
In particular, $s_\theta $ is invariant under diagonal translation
$(p=q)$.

\bigskip
 Next let $\hat \theta $ be the result of rotating $\theta $ through
$180^o$ about a point on the main diagonal. Then we have
$$
s_{\hat \theta }=\varepsilon s_\theta \leqno(9.5)
$$
where $\varepsilon $ is the involution defined by
$\varphi ^sh_r\mapsto \varphi ^{1-r-s}h_r$.

\goodbreak
\par\medskip
{\it Proof}\pointir
We may assume that $\lambda $, $\mu $ are both contained in the square 
$(n^n)$. Let $\widehat \lambda $, $\widehat \mu $ be their respective complements in
this square, so that
$\widehat  \lambda _i=n-\lambda _{n+1-i}$,
$\widehat  \mu _i=n-\mu _{n+1-i}$. Then we may take
$\widehat  \theta =\widehat  \mu -\widehat  \lambda $, so that
$$
s_{\widehat  \theta }=\det\bigl(\varphi ^{\widehat  \lambda _i-i+1}
h_{\widehat  \mu _j-\widehat  \lambda _i+i+j}\bigr)
$$
which is easily seen to be equal to
$\det\bigl(\varphi ^{i- \lambda _i}
h_{\lambda _i- \mu _j-i+j}\bigr)=\varepsilon s_{\lambda /\mu }$.\cqfd

\bigskip
 Let $\theta '=\lambda '-\mu '$ be the reflection of $\theta $ in the
main diagonal. Then we have
$$
s_{\theta '}=\omega s_\theta \leqno(9.6)
$$
where $\omega $ is the involution defined by
$\varphi ^sh_r\mapsto \varphi ^{-s}e_r$ for all $r,\,s$. Equivalently,
$$
s_{\lambda /\mu }=\det\bigl(\varphi ^{-\mu _j'+j-1}
e_{\lambda '_i-\mu' _j-i+j}\bigr).\leqno(9.6')
$$

{\it Proof}\pointir
Since the matrices $H$, $E$ (over an appropriate interval of $\bboard Z$)
are unitriangular, we have 
$\det H=\det E=1$ ; and since by (9.3) they are inverses of each other, it
follows that each minor of~$H$ is equal to the complementary cofactor of
the transpose of~$E$. This leads to $(9.6')$, exactly as in the case of
classical Schur functions
($[{\rm M}_1]$, Ch.~I, (2.9)).\cqfd

\bigskip
All the other determinant formulas for Schur functions have their
analogues in the present context, and the proofs are essentially the same.
First, if 
$\lambda =(\alpha _1,\ldots,\alpha _r\barre \beta _1,\ldots, \beta _r)$ in 
Frobenius notation, we have
$$
s_\lambda =\det\bigl(s_{(\alpha _i\barre \beta _j)}\bigr)
_{1\le i,j\le r}.\leqno(9.7)
$$

{\it Proof} (\cf. $[{\rm M}_1]$, Ch.~I, \S\kern2pt 3, Ex.~9)\pointir
If $a,\,b\ge 0$, so that $(a\barre b)=(a+1,1^b)$, the definition
$(9.1')$ gives
$$
s_{(a\barre b)}=\sum _{i=0}^b (-1)^i
\varphi ^{-i}(h_{a+i+1}) \varphi ^{-i-1}(e_{b-i})
$$
on expansion of the determinant along the top row. If $a<0$ (but $b\ge 0$)
we {\it define} $s_{(a\barre b)}$ by this formula : this definition gives
$s_{(a\barre b)}=0$ except when $a=-1-b$, in which 
$s_{(a\barre b)}=(-1)^b$.

Suppose that $l(\lambda )\le n$ and consider the $n\times n$ matrices $A$,
$B$ where
$$
A_{ij}=\varphi ^{1-j}(h_{\lambda _i-i+j}),\qquad
B_{jk}=(-1)^{j-1}\varphi ^{-j}(e_{n+1-j-k}).
$$
We have then $(AB)_{ik}=s_{(\lambda _i-i\barre n-k)}$
for $1\le i,k\le n$ ; and since $\det A=s_\lambda $ and $\det B=1$, it
follows that
$$
s_\lambda =\det\bigl(s_{(\lambda _i-i\barre n-k)}\bigr)_{1\le i,k\le n},
$$
which reduces to 
$\det\bigl(s_{(\alpha _i\barre \beta _j)}\bigr)_{1\le i,j\le r}$
exactly as in {\it loc. cit.}\cqfd

\bigskip
Next we consider the generalization of (9.7) to skew functions 
$s_{\lambda/\mu }$. Let 
$\lambda =(\alpha _1,\ldots,\alpha _r\barre \beta _1,\ldots,\beta _r)$ 
as before, and let
$\mu =(\gamma  _1,\ldots,\gamma _s\barre \varepsilon  _1,\ldots \varepsilon  _s)$.
Then we have 
$$
s_{\lambda /\mu }=(-1)^s\det\pmatrix{A&B\cr
C&0\cr}\leqno(9.8)
$$
where $A=\bigl(s_{(\alpha _i\barre \beta _j)}\bigr)_{1\le i,j\le r}$,\qquad
$B=\bigl(\varphi ^{\gamma _j+1}
h_{\alpha _i-\gamma _j}\bigr)_{1\le i\le r, 1\le j\le s}$,\hfil\break
$C=\bigl(\varphi ^{-\varepsilon _i-1}
e _{\beta _j-\varepsilon _i}\bigr)_{1\le i\le s, 1\le j\le r}$.

\medskip
The proof of (9.8) is the same as that of Lascoux and Pragacz
[${\rm LP}_1$] for classical Schur functions. (Observe that
$\varphi ^{\gamma _j+1}
h_{\alpha _i-\gamma _j}=s_{\theta _{ij}}$,
where $\theta _{ij}=(\alpha _i\barre 0)-(\gamma _j\barre 0)$, and
likewise that
$\varphi ^{-\varepsilon _i-1}
e _{\beta _j-\varepsilon _i}=s_{\varphi _{ij}}$,
where $s_{\varphi _{ij}}=(0\barre \beta _j)-(0\barre \varepsilon _i)$.)

\medskip
Finally we have, with the notation of (0.5)
$$
s_\lambda =\det\bigl(s_{[\alpha _i\barre \beta _j]}\bigr)_{1\le i,j\le r}
\leqno(9.9)
$$
exactly as in the classical case. Again, the proof is the same as in 
[${\rm LP}_2$] : indeed, both proofs given there apply in the present
context, and we therefore omit the details. There is also a ``skew" version
of (9.9) in [${\rm LP}_2$] which is likewise valid in the present context :
but it is rather complicated to state in full generality, and we shall
therefore leave its precise formulation to the conscientious reader.

\vfill\eject
\vglue 60pt
\eightpoint
\centerline{REFERENCES}
\nobreak
\vskip 10pt
\article ${\rm BL}_1$|\pd BIEDENHARN (L.C) and \pd LOUCK (J.D.)|A new
class of symmetric polynomials defined in terms of tableaux|Advances in
Appl. Math.|10|1989|396--438|
\article ${\rm BL}_2$|\pd BIEDENHARN (L.C) and \pd LOUCK
(J.D.)|Inhomogeneous basis set of symmetric polynomials defined by
tableaux|Proc. Nat. Acad. Sci. U.S.A.|87|1990|1441--1445| 
\divers CL|\pd
CHEN (W.Y.C.) and \pd LOUCK (J.D.)|The factorial Schur function, preprint (Los
Alamos, {\oldstyle 1991})| 
\article ${\rm LP}_1$|\pd LASCOUX (A.) and \pd
PRAGACZ (P.)|\'Equerres et fonctions de Schur|C.R. Acad. Sci. Paris, {\rm
S\'erie~I}|299|1984|955--958| 
\article ${\rm LP}_2$|\pd LASCOUX (A.) and
\pd PRAGACZ (P.)|Ribbon Schur functions|Europ. J.
Combinatorics|9|1988|561--574| 
\livre ${\rm M}_1$|\pd MACDONALD
(I.G.)|Symmetric functions and Hall polynomials|Oxford University Press,
$\oldstyle 1979$|  
\divers ${\rm M}_2$|\pd MACDONALD (I.G.)|A new class of
symmetric functions, Actes 20\emini~S\'eminaire Lotharingien ({\it Publ.
I.R.M.A.  Strasbourg}, $\oldstyle 1988$, 372/S--20), p.~131--171|
\article S|\pd STANLEY (R.)|Some
combinatorial properties of Jack symmetric functions|Advances in
Math.|77|1989|76--115|
\article W|\pd WACHS (M.)|Flagged Schur functions, Schubert polynomials
and symmetrizing operators|J. Comb. Theory 
{\rm Series~A}|40|1985|276--289| 

\tenpoint
\vskip 1.5cm
\rightline{\vbox{\halign{#\hfil\cr
I. G. {\petcap Macdonald},\cr
School of Mathematical Sciences,\cr
Queen Mary and Westfield College,\cr
University of London,\cr
Mile End Road,\cr
London E1 4NS,\cr
United Kingdom.\cr}}\quad}


\bye

