Character determines representation in characteristic zero

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Statement

Suppose UNIQ937dfffb724c1c04-math-00000000-QINU is a finite group and UNIQ937dfffb724c1c04-math-00000001-QINU is a field of characteristic zero. Then, the character of any finite-dimensional representation of UNIQ937dfffb724c1c04-math-00000002-QINU over UNIQ937dfffb724c1c04-math-00000003-QINU completely determines the representation, i.e., no two inequivalent finite-dimensional representations can have the same character.

Note that UNIQ937dfffb724c1c04-math-00000004-QINU does not need to be a splitting field.

Related facts

Opposite facts

Applications

Facts used

  1. Character orthogonality theorem
  2. Maschke's averaging lemma
  3. Orthogonal projection formula

Proof

Given: A group UNIQ937dfffb724c1c04-math-00000006-QINU, two linear representations UNIQ937dfffb724c1c04-math-00000007-QINU of UNIQ937dfffb724c1c04-math-00000008-QINU with the same character UNIQ937dfffb724c1c04-math-00000009-QINU over a field UNIQ937dfffb724c1c04-math-0000000A-QINU of characteristic zero.

To prove: UNIQ937dfffb724c1c04-math-0000000B-QINU and UNIQ937dfffb724c1c04-math-0000000C-QINU are equivalent as linear representations.

Proof: By Fact (2), both UNIQ937dfffb724c1c04-math-0000000D-QINU and UNIQ937dfffb724c1c04-math-0000000E-QINU are completely reducible, and are expressible as sums of irreducible representations. Suppose UNIQ937dfffb724c1c04-math-0000000F-QINU is a collection of distinct irreducible representations obtained as the union of all the representations occurring in a decomposition of UNIQ937dfffb724c1c04-math-00000010-QINU into irreducible representations and a decomposition of UNIQ937dfffb724c1c04-math-00000011-QINU into irreducible representations. In other words, there are nonnegative integers UNIQ937dfffb724c1c04-math-00000012-QINU such that:

UNIQ937dfffb724c1c04-math-00000013-QINU

and

UNIQ937dfffb724c1c04-math-00000014-QINU

Let UNIQ937dfffb724c1c04-math-00000015-QINU denote the character of UNIQ937dfffb724c1c04-math-00000016-QINU and denote by UNIQ937dfffb724c1c04-math-00000017-QINU the value UNIQ937dfffb724c1c04-math-00000018-QINU (note: this would be 1 if UNIQ937dfffb724c1c04-math-00000019-QINU were a splitting field, and in general it is the sum of squares of multiplicities of irreducible constituents over a splitting field).

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 UNIQ937dfffb724c1c04-math-0000001A-QINU and UNIQ937dfffb724c1c04-math-0000001B-QINU for each UNIQ937dfffb724c1c04-math-0000001C-QINU Fact (3) Direct application of fact
2 UNIQ937dfffb724c1c04-math-0000001D-QINU and UNIQ937dfffb724c1c04-math-0000001E-QINU for each UNIQ937dfffb724c1c04-math-0000001F-QINU UNIQ937dfffb724c1c04-math-00000020-QINU has characteristic zero, so the manipulation makes sense Step (1)
3 UNIQ937dfffb724c1c04-math-00000021-QINU for each UNIQ937dfffb724c1c04-math-00000022-QINU UNIQ937dfffb724c1c04-math-00000023-QINU has characteristic zero Step (2) [SHOW MORE]
4 UNIQ937dfffb724c1c04-math-00000028-QINU and UNIQ937dfffb724c1c04-math-00000029-QINU are equivalent Step (3)