First and second fundamental theorems of invariant theory

<h2 id="first-fundamental-theorem-for---------------------gl------------------------v----------------------displaystyle-operatorname-gl-v">First fundamental theorem for 
  
    
      
        GL
        ⁡
        (
        V
        )
      
    
    {\displaystyle \operatorname {GL} (V)}
  
</h2>
<p>The theorem states that the <a href="/facts/Ring_of_polynomial_functions/q99CvhPj">ring of 
  
    
      
        GL
        ⁡
        (
        V
        )
      
    
    {\displaystyle \operatorname {GL} (V)}
  
-invariant polynomial functions</a> on 
  
    
      
        
          
            
              V
              
                ∗
              
            
          
          
            p
          
        
        ⊕
        
          V
          
            q
          
        
      
    
    {\displaystyle {V^{*}}^{p}\oplus V^{q}}
  
 is generated by the functions 
  
    
      
        ⟨
        
          α
          
            i
          
        
        
          |
        
        
          v
          
            j
          
        
        ⟩
      
    
    {\displaystyle \langle \alpha _{i}|v_{j}\rangle }
  
, where 
  
    
      
        
          α
          
            i
          
        
      
    
    {\displaystyle \alpha _{i}}
  
 are in 
  
    
      
        
          V
          
            ∗
          
        
      
    
    {\displaystyle V^{*}}
  
 and 
  
    
      
        
          v
          
            j
          
        
        ∈
        V
      
    
    {\displaystyle v_{j}\in V}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="second-fundamental-theorem-for-general-linear-group">Second fundamental theorem for general linear group</h2>
<p>Let <i>V</i>, <i>W</i> be <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> over the complex numbers. Then the only 
  
    
      
        GL
        ⁡
        (
        V
        )
        ×
        GL
        ⁡
        (
        W
        )
      
    
    {\displaystyle \operatorname {GL} (V)\times \operatorname {GL} (W)}
  
-invariant <a href="/facts/Prime_ideal/pmrGf6DA">prime ideals</a> in 
  
    
      
        
          C
        
        [
        hom
        ⁡
        (
        V
        ,
        W
        )
        ]
      
    
    {\displaystyle \mathbb {C} [\operatorname {hom} (V,W)]}
  
 are the determinant ideal

I
          
            k
          
        
        =
        
          C
        
        [
        hom
        ⁡
        (
        V
        ,
        W
        )
        ]
        
          D
          
            k
          
        
      
    
    {\displaystyle I_{k}=\mathbb {C} [\operatorname {hom} (V,W)]D_{k}}

generated by the <a href="/facts/Determinant/eGYqJ2TF">determinants</a> of all the 
  
    
      
        k
        ×
        k
      
    
    {\displaystyle k\times k}
  
-<a href="/facts/Minor_(linear_algebra)/CMudF5Eh">minors</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Claudio_Procesi/327Bhv6R">Procesi, Claudio</a> (2007). <i>Lie groups : an approach through invariants and representations</i>. New York: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-26040-2. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/191464530">191464530</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Ch. II, § 4. of E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, <i>Geometry of algebraic curves.</i> Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985. MR0770932</li>
<li>Artin, Michael (1999). <a href="http://math.mit.edu/~etingof/artinnotes.pdf">"Noncommutative Rings"</a> (PDF).</li>
<li><a href="/facts/William_Fulton_(mathematician)/8ubLHaPy">Fulton, William</a>; <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Harris, Joe</a> (1991). <i>Representation theory. A first course</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0979-9">10.1007/978-1-4612-0979-9</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97495-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1153249">1153249</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/246650103">246650103</a>.</li>
<li>Hanspeter Kraft and Claudio Procesi, <i><a href="http://www.math.iitb.ac.in/~shripad/Wilberd/KP-Primer">Classical Invariant Theory, a Primer</a></i></li>
<li><a href="/facts/Hermann_Weyl/4C5HhyGU">Weyl, Hermann</a> (1939), <a href="https://books.google.com/books?isbn=0691057567"><i>The Classical Groups. Their Invariants and Representations</i></a>, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-05756-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0000255">0000255</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Procesi 2007, Ch. 9, § 1.4. - Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 978-0-387-26040-2. OCLC 191464530. <a href="https://search.worldcat.org/oclc/191464530" target="_blank">https://search.worldcat.org/oclc/191464530</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Procesi 2007, Ch. 13 develops this theory. - Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 978-0-387-26040-2. OCLC 191464530. <a href="https://search.worldcat.org/oclc/191464530" target="_blank">https://search.worldcat.org/oclc/191464530</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Procesi 2007, Ch. 9, § 1.4. - Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 978-0-387-26040-2. OCLC 191464530. <a href="https://search.worldcat.org/oclc/191464530" target="_blank">https://search.worldcat.org/oclc/191464530</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Procesi 2007, Ch. 11, § 5.1. - Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 978-0-387-26040-2. OCLC 191464530. <a href="https://search.worldcat.org/oclc/191464530" target="_blank">https://search.worldcat.org/oclc/191464530</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

First and second fundamental theorems of invariant theory open-in-new

First and second fundamental theorems of invariant theory