Let
M
n
(
F
)
{M_n}(F)
be an
n
×
n
n \times n
matrix ring with entries in the field F, and let
S
k
(
X
1
,
…
,
X
k
)
{S_k}({X_1}, \ldots ,{X_k})
be the standard polynomial in k variables. Amitsur-Levitzki have shown that
S
2
n
(
X
1
,
…
,
X
2
n
)
{S_{2n}}({X_1}, \ldots ,{X_{2n}})
vanishes for all specializations of
X
1
,
…
,
X
2
n
{X_1}, \ldots ,{X_{2n}}
to elements of
M
n
(
F
)
{M_n}(F)
. Now, with respect to the transpose, let
M
n
−
(
F
)
M_n^ - (F)
be the set of antisymmetric elements and let
M
n
+
(
F
)
M_n^ + (F)
be the set of symmetric elements. Kostant has shown using Lie group theory that for n even
S
2
n
−
2
(
X
1
,
…
,
X
2
n
−
2
)
{S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})
vanishes for all specializations of
X
1
,
…
,
X
2
n
−
2
{X_1}, \ldots ,{X_{2n - 2}}
to elements of
M
n
−
(
F
)
M_n^ - (F)
. By strictly elementary methods we have obtained the following strengthening of Kostant’s theorem:
S
2
n
−
2
(
X
1
,
…
,
X
2
n
−
2
)
{S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})
vanishes for all specializations of
X
1
,
…
,
X
2
n
−
2
{X_1}, \ldots ,{X_{2n - 2}}
to elements of
M
n
−
(
F
)
M_n^ - (F)
, for all n.
S
2
n
−
1
(
X
1
,
…
,
X
2
n
−
1
)
{S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}})
vanishes for all specializations of
X
1
,
…
,
X
2
n
−
2
{X_1}, \ldots ,{X_{2n - 2}}
to elements of
M
n
−
(
F
)
M_n^ - (F)
and of
X
2
n
−
1
{X_{2n - 1}}
to an element of
M
n
+
(
F
)
M_n^ + (F)
, for all n.
S
2
n
−
2
(
X
1
,
…
,
X
2
n
−
2
)
{S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})
vanishes for all specializations of
X
1
,
…
,
X
2
n
−
3
{X_1}, \ldots ,{X_{2n - 3}}
to elements of
M
n
−
(
F
)
M_n^ - (F)
and of
X
2
n
−
2
{X_{2n - 2}}
to an element of
M
n
+
(
F
)
M_n^ + (F)
, for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.