representation of orthogonal group

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Representation of the classical groups

Linear representations (cf. Linear representation ) of the groups $ \mathop{\rm GL} ( V) $, $ \textrm{SL} ( V) $, $ \textrm{O} ( V, f ) $, $ \textrm{SO} ( V, f ) $, $ \textrm{Sp} ( V, f ) $, where $ V $ is an $ n $-dimensional vector space over a field $ k $ and $ f $ is a non-degenerate symmetric or alternating bilinear form on $ V $, in invariant subspaces of tensor powers $ T ^ {m} ( V) $ of $ V $. If the characteristic of $ k $ is zero, then all irreducible polynomial linear representations of these groups can be realized by means of tensors.

In the case $ k = \mathbf C $ the groups above are complex Lie groups. For all groups, except $ \textrm{GL} ( V) $, all (differentiable) linear representations are polynomial; every linear representation of $ \textrm{ GL} ( V) $ has the form $ g \mapsto ( \det g) ^ {k} R ( g) $, where $ k \in \mathbf Z $ and $ R $ is a polynomial linear representation. The classical compact Lie groups $ \textrm{U} _ {n} $, $ \textrm{SU} _ {n} $, $ \textrm{O} _ {n} $, $ \textrm{SO} _ {n} $, and $ \textrm{Sp} _ {n} $ have the same complex linear representations and the same invariant subspaces in tensor spaces as their complex envelopes $ \textrm{U} _ {n} ( \mathbf C ) $, $ \textrm{SL} _ {n} ( \mathbf C ) $, $ \textrm{O} _ {n} ( \mathbf C ) $, $ \textrm{SO} _ {n} ( \mathbf C ) $, and $ \textrm{Sp} _ {n} ( \mathbf C ) $. Therefore, results of the theory of linear representations obtained for the classical complex Lie groups can be carried over to the corresponding compact groups and vice versa (Weyl's "unitary trick" ). In particular, using integration on a compact group one can prove that linear representations of the classical complex Lie groups are completely reducible.

The natural linear representation of $ \textrm{GL} ( V) $ in $ T ^ {m} ( V) $ is given by the formula

$$ g ( v _ {1} \otimes \dots \otimes v _ {m} ) = \ gv _ {1} \otimes \dots \otimes gv _ {m} ,\ \ g \in \textrm{GL} ( V),\ \ v _ {i} \in V. $$

In the same space a linear representation of the symmetric group $ S _ {m} $ is defined by

$$ \sigma ( v _ {1} \otimes \dots \otimes v _ {m} ) = \ v _ {\sigma ^ {- 1 }( 1) } \otimes \dots \otimes v _ {\sigma ^ {- 1 }( m) } ,\ \ \sigma \in S _ {m} ,\ \ v _ {i} \in V. $$

The operators of these two representations commute, so that a linear representation of $ \textrm{GL} ( V) \times S _ {m} $ is defined in $ T ^ {m} ( V) $. If $ \mathop{\rm char} k = 0 $, the space $ T ^ {m} ( V) $ can be decomposed into a direct sum of minimal $ ( \textrm{GL} ( V) \times S _ {m} ) $-invariant subspaces:

$$ T ^ {m} ( V) = \ \sum _ \lambda V _ \lambda \otimes U _ \lambda . $$

The summation is over all partitions $ \lambda $ of $ m $ containing at most $ n $ summands, $ U _ \lambda $ is the space of the absolutely-irreducible representation $ T _ \lambda $ of $ S _ {m} $ corresponding to $ \lambda $ (cf. Representation of the symmetric groups ) and $ V _ \lambda $ is the space of an absolutely-irreducible representation $ R _ \lambda $ of $ \textrm{GL} ( V) $. A partition $ \lambda $ can be conveniently represented by a tuple $ ( \lambda _ {1} \dots \lambda _ {n} ) $ of non-negative integers satisfying $ \lambda _ {1} \geq \dots \geq \lambda _ {n} $ and $ \sum _ {i} \lambda _ {i} = m $.

The subspace $ V _ \lambda \otimes U _ \lambda \subset T ^ {m} ( V) $ splits in a sum of minimal $ \textrm{GL} ( V) $-invariant subspaces, in each of which a representation $ R _ \lambda $ can be realized. These subspaces can be explicitly obtained by using Young symmetrizers (cf. Young symmetrizer ) connected with $ \lambda $. E.g. for $ \lambda = ( m, 0 \dots 0) $ (respectively, $ \lambda = ( 1 \dots 1, 0 \dots 0) $ for $ m \leq n $) one has $ \dim U _ \lambda = 1 $ and $ V _ \lambda \otimes U _ \lambda $ is the minimal $ \textrm{GL} ( V) $-invariant subspace consisting of all symmetric (respectively, skew-symmetric) tensors.

The representation $ R _ \lambda $ is characterized by the following properties. Let $ B \subset \textrm{GL} ( V) $ be the subgroup of all linear operators that, in some basis $ \{ e _ {1} \dots e _ {n} \} $ of $ V $, can be written as upper-triangular matrices. Then the operators $ R _ \lambda ( b) $, $ b \in B $, have a unique (up to a numerical factor) common eigenvector $ v _ \lambda $, which is called the highest weight vector of $ R _ \lambda $. The corresponding eigenvalue (the highest weight of $ R _ \lambda $) is equal to $ b _ {11} ^ {\lambda _ {1} } \dots b _ {nn} ^ {\lambda _ {n} } $, where $ b _ {ii} $ is the $ i $-th diagonal element of the matrix of $ b $ in the basis $ \{ e _ {1} \dots e _ {n} \} $. Representations $ R _ \lambda $ corresponding to distinct partitions $ \lambda $ are inequivalent. The character of $ R _ \lambda $ can be found from Weyl's formula

$$ \tr R _ \lambda ( g) = \ \frac{W _ \lambda ( z _ {1} \dots z _ {n} ) }{W _ {0} ( z _ {1} \dots z _ {n} ) } , $$

where $ z _ {1} \dots z _ {n} $ are the roots of the characteristic polynomial of the operator $ g $, $ W _ \lambda $ is the generalized Vandermonde determinant corresponding to $ \lambda $ (cf. Frobenius formula ) and $ W _ {0} $ is the ordinary Vandermonde determinant. The dimension of $ R _ \lambda $ is equal to

$$ \dim R _ \lambda = \ \prod _ {i < j } \frac{l _ {i} - l _ {j} }{j - i } , $$

where $ l _ {i} = \lambda _ {i} + n - i $.

The restriction of $ R _ \lambda $ to the unimodular group $ \textrm{SL} ( V) $ is irreducible. The restrictions to $ \textrm{SL} ( V) $ of two representations $ R _ \lambda $ and $ R _ \mu $ are equivalent if and only if $ \mu _ {i} = \lambda _ {i} + s $ (where $ s $ is independent of $ i $). The restriction of a representation $ R _ \lambda $ of $ \textrm{GL} _ {n} ( k) $ to the subgroup $ \textrm{GL} _ {n - 1 } ( k) $ can be found by the rule:

$$ R _ \lambda \mid _ { \textrm{GL} _ {n - 1 } ( k) } = \ \sum _ \mu R _ \mu , $$

where $ \mu $ runs through all tuples $ ( \mu _ {1} \dots \mu _ {n - 1 } ) $ satisfying

$$ \lambda _ {1} \geq \mu _ {1} \geq \ \lambda _ {2} \geq \mu _ {2} \geq \dots \geq \lambda _ {n} . $$

For every Young diagram $ d $, corresponding to a partition $ \lambda $, the tensor $ v _ \lambda \otimes u _ {d} ^ \prime \in T ^ {m} ( V) $ (for notations see Representation of the symmetric groups ) is the result of alternating the tensor $ e _ {i _ {1} } \otimes \dots \otimes e _ {i _ {m} } $ over the columns of $ d $, where $ i _ {k} $ is the number of the row of $ d $ in which the number $ k $ is located. The tensors thus constructed with respect to all standard diagrams $ d $ form a basis of the minimal $ S _ {m} $-invariant subspace of $ v _ \lambda \otimes U _ \lambda $ in which the representation $ T _ \lambda $ of $ S _ {m} $ is realized.

A linear representation of the orthogonal group $ \textrm{O} ( V, f ) $ in $ T ^ {m} ( V) $ has the following structure. There is a decomposition into a direct sum of two $ ( \textrm{O} ( V, f ) \times S _ {m} ) $-invariant subspaces:

$$ T ^ {m} ( V) = T _ {0} ^ {m} ( V) \oplus T _ {1} ^ {m} ( V) , $$

where $ T _ {0} ^ {m} ( V) $ consists of traceless tensors, i.e. tensors whose convolution with $ f $ over any two indices vanishes, and

$$ T _ {1} ^ {m} ( V) = \ \sum _ {\sigma \in S _ {m} } \sigma ( T ^ {m - 2 } ( V) \otimes f ^ { - 1 } ). $$

The space $ T _ {0} ^ {m} ( V) $, in turn, decomposes into a direct sum of $ ( \textrm{O} ( V, f ) \times S _ {m} ) $-invariant subspaces:

$$ T _ {0} ^ {m} ( V) = \ \sum _ \lambda V _ \lambda ^ {0} \otimes U _ \lambda , $$

where $ V _ \lambda ^ {0} \subset V _ \lambda $. Moreover, $ V _ \lambda ^ {0} \neq 0 $ if and only if the sum $ \lambda _ {1} ^ \prime + \lambda _ {2} ^ \prime $ of the heights of the first two columns of the Young tableau corresponding to $ \lambda $ does not exceed $ n $, and in this case $ V _ \lambda ^ {0} $ is the space of an absolutely-irreducible representation $ R _ \lambda ^ {0} $ of $ \textrm{O} ( V, f ) $. Representations $ R _ \lambda ^ {0} $ corresponding to distinct partitions $ \lambda $ are inequivalent. If $ \lambda $ satisfies the condition $ \lambda _ {1} ^ \prime + \lambda _ {2} ^ \prime \leq n $, then after replacing the first column of its Young tableau by a column of height $ n - \lambda _ {1} ^ \prime $ one obtains the Young tableau of a partition $ \overline \lambda $ which also satisfies this condition. The corresponding representations of $ \textrm{O} ( V, f ) $ are related by $ R _ {\overline \lambda } ^ {0} ( g) = ( \det g) R _ \lambda ^ {0} ( g) $ (in particular, they have equal dimension).

The restriction of $ R _ \lambda ^ {0} $ to the subgroup $ \textrm{SO} ( V, f ) $ is absolutely irreducible, except in the case $ n $ even and $ \lambda = \overline \lambda $ (i.e. the number of terms of $ \lambda $ is equal to $ n/2 $). In the latter case it splits over the field $ k $ or a quadratic extension of it into a sum of two inequivalent absolutely irreducible representations of equal dimension.

In computing the dimension of $ R _ \lambda ^ {0} $ one can assume that $ \lambda _ {1} ^ \prime \leq n/2 $ (otherwise replace $ \lambda $ by $ \overline \lambda $). Let $ l _ {i} = \lambda _ {i} + n/2 - i $. Then for odd $ n $ one has

$$ \dim R _ \lambda ^ {0} = \ \prod _ {i = 1 } ^ { [ n/2]} \frac{l _ {i} }{n/2 - i } \prod _ {\begin{array}{c} i, j = 1 \\ i < j \end{array} } ^ { [ n/2]} \frac{( l _ {i} - l _ {j} ) ( l _ {i} + l _ {j} ) }{( j - i) ( n - i - j) } , $$

while for even $ n $ and $ \lambda \neq \overline \lambda $ one has

$$ \dim R _ \lambda ^ {0} = \ \prod _ {\begin{array}{c} i, j = 1 \\ i < j \end{array} } ^ { n/2 } \frac{( l _ {i} - l _ {j} ) ( l _ {i} + l _ {j} ) }{( j - i) ( n - i - j) } . $$

For $ \lambda = \overline \lambda $ the latter formula gives half the dimension of $ R _ \lambda ^ {0} $, i.e. the dimension of each of the absolutely-irreducible representations of $ \textrm{SO} ( V, f ) $ corresponding to it.

The decomposition of $ T ^ {m} ( V) $ with respect to the symplectic group $ \textrm{Sp} ( V, f ) $ is analogous to the decomposition with respect to the orthogonal group, with the difference that $ V _ \lambda ^ {0} \neq 0 $ if and only if $ \lambda _ {1} ^ \prime \leq n/2 $. The dimension of $ R _ \lambda ^ {0} $ can in this case be found from

$$ \dim R _ \lambda ^ {0} = \ \prod _ {i = 1 } ^ { n/2 } \frac{l _ {i} }{n/2 - i + 1 } \prod _ {\begin{array}{c} i, j = 1 \\ i < j \end{array} } ^ { n/2 } \frac{( l _ {i} - l _ {j} ) ( l _ {i} + l _ {j} ) }{( j - i) ( n - j - i + 2) } , $$

where $ l _ {i} = \lambda _ {i} - i + 1 + n/2 $.

This article describes the classical theory. The contemporary period in this old field of algebra began with [a1] . It can be described by two words: "characteristic-free representation theory" . A different approach to the polynomial representations of $ \textrm{GL} ( V) $ and $ \textrm{SL} ( V) $ was undertaken in [a2] . Further, both classical and characteristic free theories can be found in [a3] .

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Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field

Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two. I am looking for a reference that explains how to describe the simple (finite-dimensional) representations of $O(n)$ in terms of the simple representations of $SO(n)$.

The relation between the complex representations of the corresponding compact Lie groups is explained in section VI.7 of Bröcker and tom Dieck’s Representations of Compact Lie Groups . I believe the key statements made there also hold in the above algebraic situation, and I have worked this out in a fair amount of detail, but I am currently unwilling to believe that this has not already been done somewhere in the literature. Roughly, these statements are as follows:

When $n$ is odd, $O(n)$ is a direct product of $SO(n)$ with $\mathbb Z/2$. Thus, every simple $SO(n)$-representation can be lifted to two distinct $O(n)$-representations, and every $O(n)$-representation arises in this way. (See also this question regarding simple representations of products .)

When $n$ is even, $O(n)$ is only a semi-direct product of $SO(n)$ with $\mathbb Z/2$. In this case, only some simple $SO(n)$-representations can be lifted. Those that can be lifted can again be lifted to two distinct $O(n)$-representations. The remaining simple $SO(n)$-representations occur in pairs whose direct sum can be lifted to a unique simple $O(n)$-representation. All simple $O(n)$-representations arise in either of these ways.

Of course, in this question I am mainly interested in references concerning the case when $n$ is even.

• reference-request
• rt.representation-theory
• algebraic-groups

• $\begingroup$ For the time being, the “fair amount of detail” I am referring to may be found in Proposition 3.18 and in Section 4.2 of arXiv:1308.0796 . $\endgroup$ –  Communicative Algebra Commented Aug 7, 2013 at 7:49
• 4 $\begingroup$ For $k=\mathbb C$, it's explained in Section 5.5.5 of Goodman-Wallach's book "Symmetry, represenations, and invariants". $\endgroup$ –  emiliocba Commented Aug 13, 2013 at 22:17
• $\begingroup$ As long as the characteristic of the field of definition is good (not 2), as you assume, it doesn't seem to matter over which field the groups are defined and split. Representations will be studied over an algebraically closed field, where for $n$ even the methods will rely on standard induction/restriction. Older group representation texts for physicists probably cover orthogonal groups, while Jantzen's book on algebraic groups may be overkill. $\endgroup$ –  Jim Humphreys Commented Aug 14, 2013 at 17:34
• $\begingroup$ P.S. As pointed out in a comment, the Springer GTM 255 by Goodman-Wallach is likely to be a reliable source for your purpose. I don't have that 2009 edition, but the first edition titled Representations and Invariants of the Classical Groups (Cambridge, 1998) has the relevant material in sections 5.2.2 and 10.2.5. $\endgroup$ –  Jim Humphreys Commented Aug 14, 2013 at 19:00
• $\begingroup$ @JimHumphreys: I’m not sure what you mean by “representations will be studied over an algebraically closed field”—I want the representations to be defined over the same field as the group itself. Do I need to make this more precise in the question? $\endgroup$ –  Communicative Algebra Commented Aug 15, 2013 at 7:48

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Irreducible Representation

1. The dimensionality theorem :

3. Orthogonality of different representations

4. In a given representation, reducible or irreducible, the group characters of all matrices belonging to operations in the same class are identical (but differ from those in other representations).

6. A one-dimensional representation with all 1s (totally symmetric) will always exist for any group .

7. A one-dimensional representation for a group with elements expressed as matrices can be found by taking the group characters of the matrices .

Irreducible representations can be indicated using Mulliken symbols .

Portions of this entry contributed by Todd Rowland

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Rowland, Todd and Weisstein, Eric W. "Irreducible Representation." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/IrreducibleRepresentation.html

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Confused by "orthogonal representation" of a finite group. How to show the matrix under some basis from the new defined inner product is orthogonal?

Suppose we have a finite group $G$ and a representation $\rho:G\to GL(V)$ on the finite vector space $V$ . So $V$ is a $G$ -module.

Now I am asked to show that there is a basis of $V$ under which the matrix of $\rho(g)$ is orthogonal for all $g\in G$ . Here is what confused me and I still don't know in what sense is this possible. I have done some searching and know typically the steps are the following:(here we assume $\rho(g)$ is the matrix under standard basis)

(1) Define a $G$ -invariant inner product $[v,w]=\sum_{g\in G}\langle\rho(g)v,\rho(g)w\rangle$ , where $\langle,\rangle$ is the usual inner product under standard basis. (2) Choose a basis of $V$ that is orthogonal w.r.t this inner product. (3) Under such basis, the matrix of $\rho(g)$ is orthogonal for all $g\in G$ .

Questions: does the (2) mean we find a matrix $B$ s.t. $B^H\sum_{g\in G}\rho(g)^H\rho(g)B=I$ ? If yes, then the matrix of $\rho(g)$ under the basis $B$ should become $M(g)=B^{-1}\rho(g)B$ right? But how could we see $M(g)^HM(g)=I$ ?

I guess I must be misunderstanding somewhere. Any help would be appreciated. Thanks.

Update: I think I got this. I was stupid. I think in this way: if let $\sum_{g\in G}\rho(g)^H\rho(g)=Q^HQ$ positive definiteness, then for each $g\in G$ we have $\rho(g)^HQ^HQ\rho(g)=Q^HQ$ from $G$ -invariance. Then this shows $Q\rho(g)Q^{-1}$ is unitary(orthogonal). Now let $B=Q^{-1}$ then we are done. This is is the same as the one in (2).

• abstract-algebra
• representation-theory

2 Answers 2

I think I'm misunderstanding your question, but (2) means we find vectors $\{w_i\} \subset V$ such that $[w_i, w_j] = \delta_{ij}$ . In other words, $[w_i, w_j] = 0$ for $i \neq j$ and $[w_i, w_i] = 1$ . This is done with the Gram-Schmidt process .

Update : I'm having trouble understanding your notation and the matrix $B$ . But maybe if I put (3) in my notation then you can translate?

Note that by $G$ -invariance, for all $w_1, w_2 \in V$ we have $$[\rho(g)w_1, w_2] = [w_1, \rho(g^{-1})w_2] = [w_1, \rho(g)^{-1}w_2].$$ Moreover, the Hermitian transpose satisfies $$[\rho(g)w_1, w_2] = [w_1, \rho(g)^H w_2]$$ In other words $$[w_1, \rho(g)^{-1}w_2] = [w_1, \rho(g)^H w_2].$$

Now plugging in the $G$ -invariant basis from (2), you can get the equality of the matrices $\rho(g)^{-1}$ and $\rho(g)^H$

• $\begingroup$ I think this is the same as what I describe in (2) if you treat $B$ consists of columns of your $w_i,w_j$. $\endgroup$ –  Gnon Commented Jan 28, 2021 at 0:51
• 1 $\begingroup$ I took another crack at it based on your comment -- hope it helps $\endgroup$ –  Sam Freedman Commented Jan 28, 2021 at 1:17
• $\begingroup$ Thank you. I think I know it now. $\endgroup$ –  Gnon Commented Jan 28, 2021 at 2:09

Let $\gamma\in G$ . Then for all $v$ , $w\in V$ one has $$ [\rho(\gamma)v,\rho(\gamma)w]= \sum_{g\in G}\langle\rho(g\gamma)v,\rho(g\gamma)w\rangle= \sum_{g^\prime\in G}\langle\rho(g^\prime)v,\rho(g^\prime)w\rangle= [v,w] $$ because the map $g\mapsto g\gamma$ is a bijection of $G$ onto itself.

But the above equality tells exactly that $\rho(g)$ leaves $[\cdot,\cdot]$ unchanged, so that its associated matrix with respect to an orthonormal basis will be orthogonal

• $\begingroup$ Yes this is the part I understand. I always suppose this is the way we explain literally the "orthogonality" of the matrix. But I fail to really showing this in the formulaic sense like $M^HM=I$. Isn't this the mathematical definition of a orthogonal matrix(in this particular question)? $\endgroup$ –  Gnon Commented Jan 28, 2021 at 0:57
• 1 $\begingroup$ @gnon: I'm not sure about your notation $M^H$ (I assume that denotes transposition), but yes: the matrix of an orthogonal transformation has that property if written in terms of an orthonormal basis (making it an orthogonal matrix,definitions are consistent). Writing $M$ explicitly from the original matrix of $\rho$ looks cumbersome, but is not such a deep problem. $\endgroup$ –  Andrea Mori Commented Jan 28, 2021 at 1:08
• $\begingroup$ Yes you are right. I got it now. Thanks! $\endgroup$ –  Gnon Commented Jan 28, 2021 at 2:09

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Basis of irreducible representations of orthogonal groups

• Original Papers
• Published: September 1989
• Volume 7 , pages 127–139, ( 1989 )

Cite this article

• T. Babutsidze 1 ,
• I. Machabeli 1 ,
• Z. Shavtvalishvili 1 &
• I. Vashakmadze 1  

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A method for constructing a basis of irreducible representations of unitary groups U n is generalized to orthogonal groups O n . The corresponding algorithm is given for the chain for groups U n ⊃ O n and also for the chain

which is of great relevance in physics.

The constructed basis is used for the calculation of the fractional six-particle parentage coefficients for separation of one-, two- and three-particle wave functions. All possible states with excitation quanta N ≤2 are taken into account. The present results, together with previous work, are used for the investigation of the six-quark system, in particular for calculating the D -state probability in the ground state of this system.

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Institute of High-Energy Physics, Tbilisi State University, University Street 9, SU-380086, Tbilisi, USSR

T. Babutsidze, I. Machabeli, Z. Shavtvalishvili & I. Vashakmadze

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Babutsidze, T., Machabeli, I., Shavtvalishvili, Z. et al. Basis of irreducible representations of orthogonal groups. Few-Body Systems 7 , 127–139 (1989). https://doi.org/10.1007/BF01087392

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Received : 11 November 1988

Revised : 02 August 1989

Accepted : 15 September 1989

Issue Date : September 1989

DOI : https://doi.org/10.1007/BF01087392

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