Formula for calculating effect of Schur functor on character
Contents
Statement
Suppose is a group,
is a field (which we will assume for simplicity to have characteristic zero) and
is a linear representation of
over
with character
. Suppose
is a partition of a positive integer
and denote by
the Schur functor corresponding to
. We can consider a new representation
and we denote its character by
. Our goal is to provide an explicit description of
.
Explicit description of degree of representation
The new degree is a polynomial in the old degree. The degree of the polynomial (not to be confused with degrees of representations) equals . Explicitly, the polynomial is defined as follows. Denote by
the set of conjugacy classes of the symmetric group
. Let
be the function that outputs the number of cycles of a given conjugacy class. Denote by
the character of the irreducible representation of
corresponding to the partition
of
. Then, the polynomial is:
Explicit description of character of representation
The character of the representation is given as follows. Denote by the set of conjugacy classes of the symmetric group
. Let
be the function that outputs the number of cycles of a given conjugacy class. Denote by
the character of the irreducible representation of
corresponding to the partition
of
. Then, the character is:
For instance, for a conjugacy class in with cycle type
, the product on the inside reads
.
Description for small Schur functors
Size of set being partitioned | Partition for Schur functor | Name of functor | Formula for computing degree if original representation has degree ![]() |
Formula for computing character of the representation obtained after applying this functor in terms of the original character ![]() ![]() |
Explanation for character formula |
---|---|---|---|---|---|
1 | 1 | identity functor | ![]() |
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2 | 2 | symmetric square | ![]() |
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![]() ![]() ![]() partition 1 + 1 of 2, corresponding to conjugacy class of identity element: the summand is ![]() partition 2 of 2, corresponding to conjugacy class of non-identity element: the summand is ![]() Adding, we get the result. |
2 | 1 + 1 | exterior square or alternating square | ![]() |
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3 | 3 | symmetric cube | ![]() |
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3 | 2 + 1 | ? | ![]() |
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3 | 1 + 1 + 1 | exterior cube | ![]() |
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