Difference between revisions of "Contragredient representation"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.B. Vinberg, A.L. Onishchik, "Seminar on algebraic groups and Lie groups 1967/68" , Springer (Forthcoming) (Translated from Russian)</TD></TR></table> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) {{MR|0412321}} {{ZBL|0342.22001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.B. Vinberg, A.L. Onishchik, "Seminar on algebraic groups and Lie groups 1967/68" , Springer (Forthcoming) (Translated from Russian)</TD></TR></table> |
Revision as of 10:02, 24 March 2012
to a representation of a group
in a linear space
The representation of the same group
in the dual space
of
defined by the rule
![]() |
for all , where
denotes taking adjoints.
More generally, if is a linear space over the same field
as
and
is a non-degenerate bilinear form (pairing) on
with values in
, then a representation
of
in
is called the representation contragredient to
with respect to the form
if
![]() |
for all ,
,
.
For example, if is the general linear group of a finite-dimensional space
, then the natural representation of
in the space of covariant tensors of fixed rank on
is the representation contragredient to the natural representation of
in the space of contravariant tensors of the same rank on
.
Let be finite-dimensional over
, let
be a basis of it, and let
be the basis dual to
in
. Then, for any
in
, the matrix of
in the basis
is obtained from the matrix of the operator
in the basis
by taking the transpose of the inverse. If
is irreducible, then so is
. If
is a Lie group with Lie algebra
, and
and
are the representations of the algebra
induced, respectively, by two representations
and
of
in spaces
and
that are contragredient with respect to the pairing
, then
![]() | (*) |
for all ,
,
. Representations of a Lie algebra
satisfying the condition (*) are also called contragredient representations with respect to
.
Suppose further that is a complex, connected, simply-connected semi-simple Lie group and that
is an irreducible finite-dimensional representation of it in a linear space
. The weights of the representation
are opposite to those of
(see Weight of a representation of a Lie algebra), the lowest weight of
being opposite to the highest weight of
(see Cartan theorem on the highest (weight) vector). The representations
and
are equivalent if and only if there is a non-zero bilinear form on
that is invariant with respect to
. If such a form exists, then it is non-degenerate and either symmetric or skew-symmetric. The set of numerical marks of the highest weight of the representation
is obtained from the set of numerical marks of
by applying the substitution induced by the following automorphism
of the Dynkin diagram of simple roots
of
:
a) takes each connected component
,
, of
into itself;
b) if is a diagram of type
,
or
, then the restriction of
to
is uniquely defined as the unique element of order 2 in the automorphism group of
; in the remaining cases the restriction of
to
is the identity.
References
[1] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018 |
[2] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) MR0412321 Zbl 0342.22001 |
[3] | D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013 |
[4] | E.B. Vinberg, A.L. Onishchik, "Seminar on algebraic groups and Lie groups 1967/68" , Springer (Forthcoming) (Translated from Russian) |
Comments
If is the highest weight of the highest weight representation
, then the set of numerical marks of
is simply the ordered set of integers
,
; cf. Cartan theorem, especially when written as labels at the corresponding nodes of the Dynkin diagram.
Contragredient representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contragredient_representation&oldid=21833