Editing Symbolic method

{{For|the method in analytic combinatorics|Symbolic method (combinatorics)}}
{{Use shortened footnotes|date=May 2021}}  
In [[mathematics]], the '''symbolic method''' in [[invariant theory]] is an [[algorithm]] developed by [[Arthur Cayley]],{{r|Cayley1846}} [[Siegfried Heinrich Aronhold]],{{r|Aronhold1858}} [[Alfred Clebsch]],{{r|Clebsch1861}} and [[Paul Gordan]]{{sfn|Gordan|1887}} in the 19th century for computing  [[invariant (mathematics)|invariant]]s of [[algebraic form]]s. It is based on treating the form as if it were a power of a degree one form, which corresponds to embedding a symmetric power of a vector space into the symmetric elements of a [[tensor product]] of copies of it.

==Symbolic notation==
The symbolic method uses a compact, but rather confusing and mysterious notation for invariants, depending on the introduction of new symbols ''a'', ''b'', ''c'', ...  (from which the symbolic method gets its name) with apparently contradictory properties.

===Example: the discriminant of a binary quadratic form===
These symbols can be explained by the following example from Gordan.{{sfn|Gordan|1887|loc=v. 2, p.g. 1-3}} Suppose that 
:<math>\displaystyle  f(x) = A_0x_1^2+2A_1x_1x_2+A_2x_2^2</math>
is a binary quadratic form with an invariant given by the discriminant
:<math>\displaystyle \Delta=A_0A_2-A_1^2.</math>
The symbolic representation of the discriminant is 
:<math>\displaystyle 2\Delta=(ab)^2</math>
where ''a'' and ''b'' are the symbols. The meaning of  the expression (''ab'')<sup>2</sup> is as follows. First of all, (''ab'') is a shorthand form for the determinant of a matrix whose rows are ''a''<sub>1</sub>, ''a''<sub>2</sub> and ''b''<sub>1</sub>, ''b''<sub>2</sub>, so 
:<math>\displaystyle (ab)=a_1b_2-a_2b_1.</math>
Squaring this we get
:<math>\displaystyle (ab)^2=a_1^2b_2^2-2a_1a_2b_1b_2+a_2^2b_1^2.</math>
Next we pretend that
:<math>\displaystyle f(x)=(a_1x_1+a_2x_2)^2=(b_1x_1+b_2x_2)^2</math>
so that
:<math>\displaystyle  A_i=a_1^{2-i}a_2^{i}= b_1^{2-i}b_2^{i}</math>
and we ignore the fact that this does not seem to make sense if ''f'' is not a power of a linear form. 
Substituting these values gives
:<math>\displaystyle (ab)^2= A_2A_0-2A_1A_1+A_0A_2 = 2\Delta.</math>

===Higher degrees===
More generally if 
:<math>\displaystyle  f(x) = A_0x_1^n+\binom{n}{1}A_1x_1^{n-1}x_2+\cdots+A_nx_2^n</math>
is a binary form of higher degree, then one introduces new variables ''a''<sub>1</sub>, ''a''<sub>2</sub>,  ''b''<sub>1</sub>, ''b''<sub>2</sub>, ''c''<sub>1</sub>, ''c''<sub>2</sub>, with the properties
:<math>f(x)=(a_1x_1+a_2x_2)^n=(b_1x_1+b_2x_2)^n=(c_1x_1+c_2x_2)^n=\cdots.</math>

What this means is that the following two vector spaces are naturally isomorphic:
*The  vector space of homogeneous polynomials in ''A''<sub>0</sub>,...''A''<sub>''n''</sub> of degree ''m''
*The vector space of  polynomials in 2''m'' variables ''a''<sub>1</sub>, ''a''<sub>2</sub>,  ''b''<sub>1</sub>, ''b''<sub>2</sub>, ''c''<sub>1</sub>, ''c''<sub>2</sub>, ... that have degree ''n'' in each of the ''m'' pairs of variables (''a''<sub>1</sub>, ''a''<sub>2</sub>),  (''b''<sub>1</sub>, ''b''<sub>2</sub>), (''c''<sub>1</sub>, ''c''<sub>2</sub>), ... and are symmetric under permutations of the ''m'' symbols ''a'', ''b'', ....,
The isomorphism is given by mapping ''a''{{su|p=''n''&minus;''j''|b=1}}''a''{{su|p=''j''|b=2}}, ''b''{{su|p=''n''&minus;''j''|b=1}}''b''{{su|p=''j''|b=2}}, .... to ''A''<sub>''j''</sub>. This mapping does not preserve products of polynomials.

===More variables===
The extension to a form ''f'' in more than two variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>,... is similar: one introduces symbols ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> and so on  with the properties
:<math>f(x)=(a_1x_1+a_2x_2+a_3x_3+\cdots)^n=(b_1x_1+b_2x_2+b_3x_3+\cdots)^n=(c_1x_1+c_2x_2+c_3x_3+\cdots)^n=\cdots.</math>

==Symmetric products==

The rather mysterious formalism of the symbolic method corresponds to embedding a symmetric product S<sup>''n''</sup>(''V'') of a vector space ''V'' into a tensor product of ''n'' copies of ''V'', as the elements preserved by the action of the symmetric group. In fact this is done twice, because the invariants of degree ''n'' of a quantic of degree ''m'' are the invariant elements of S<sup>''n''</sup>S<sup>''m''</sup>(''V''), which gets embedded into a tensor product of ''mn'' copies of ''V'', as the elements invariant under a wreath product of the two symmetric groups. The brackets of the symbolic method are really invariant linear forms on this tensor product, which give invariants of S<sup>''n''</sup>S<sup>''m''</sup>(''V'') by restriction.

==See also==
*[[Umbral calculus]]

==References==
* {{Cite book |last=Gordan |first=Paul |date=1987 |orig-date=1887 |editor-last=Kerschensteiner |editor-first=Georg |title=Vorlesungen über Invariantentheorie |edition=2nd |location=New York York |publisher=[[American Mathematical Society|AMS]] [[Chelsea Publishing Company|Chelsea Publishing]] |isbn=9780828403283 |mr=917266 |url=https://books.google.com/books?id=d6RLAAAAMAAJ |ref=CITEREFGordan1887 }}

'''Footnotes'''
{{reflist|refs=
<ref name=Cayley1846>{{Cite journal |last=Cayley |first=Arthur |author-link=Arthur Cayley |date=1846 |title=On linear transformations | journal=Cambridge and Dublin Mathematical Journal  |pages=104–122 |url=http://resolver.sub.uni-goettingen.de/purl?PPN600493962_0001}}</ref>

<ref name=Aronhold1858>{{Cite journal |last1=Aronhold |first1=Siegfried Heinrich |date=1858 |title=Theorie der homogenen Functionen dritten Grades von drei Veränderlichen |journal=[[Journal für die reine und angewandte Mathematik]] |volume=1858 |issue=55 |pages=97–191 |issn=0075-4102 |doi=10.1515/crll.1858.55.97|s2cid=122247157 |language=de |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00215028X }}</ref>

<ref name=Clebsch1861>{{Cite journal |last1=Clebsch |first1=A. |date=1861 |title=Ueber symbolische Darstellung algebraischer Formen |journal=Journal für die Reine und Angewandte Mathematik |volume=1861 |issue=59 |pages=1–62 |issn=0075-4102 |doi=10.1515/crll.1861.59.1|s2cid=119389672 |language=de |url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0059 }}</ref>
}}

==Further reading==
* {{cite journal |last1=Dieudonné |first1=Jean |author-link1=Jean Dieudonné |last2=Carrell |first2=James B. | authorlink2=James B. Carrell |date= 1970 |title=Invariant theory, old and new |journal=[[Advances in Mathematics]] |volume=4 |pages=1–80 |doi-access=free |doi=10.1016/0001-8708(70)90015-0}} pp.&nbsp;32–7, "Invariants of ''n''-ary forms: the symbolic method. Reprinted as {{cite book |last1=Dieudonné |first1=Jean |author-link=Jean Dieudonné |last2=Carrell |date=1971 |first2=James B. |title=Invariant theory, old and new |publisher=Academic Press |isbn=0-12-215540-8}}
* {{Cite book |last1=Dolgachev |first1=Igor |date=2003 |title=Lectures on invariant theory |volume=296 |series=London Mathematical Society Lecture Note Series |publisher=[[Cambridge University Press]] |isbn=978-0-521-52548-0 |doi=10.1017/CBO9780511615436 |mr=2004511 |s2cid=118144995 }}
* {{citation |last1= Grace |first1=John Hilton |author-link=John Hilton Grace |last2=Young |first2=Alfred |author-link2=Alfred Young  (mathematician) |date=1903 |title=The Algebra of invariants |publisher=Cambridge University Press}}
* {{Cite book |last1=Hilbert |first1=David |author1-link=David Hilbert |date=1993 |orig-date=1897 |title=Theory of algebraic invariants |publisher=[[Cambridge University Press]] |isbn=9780521444576 |mr=1266168 |url=https://books.google.com/books?isbn=0521449030}}
* {{cite book |editor-last=Koh |editor-first=Sebastian S. |date=2009 |orig-date=1987 |title=Invariant Theory |series=Lecture Notes in Mathematics |volume=1278 |publisher=Springer |isbn=9783540183600}}
* {{Cite journal |last1=Kung |first1=Joseph P. S. |last2=Rota |first2=Gian-Carlo |author2-link=Gian-Carlo Rota |date=1984 |title=The invariant theory of binary forms |journal=Bulletin of the American Mathematical Society |series=New Series |volume=10 |issue=1 |pages=27–85 |issn=0002-9904 |doi-access=free |doi=10.1090/S0273-0979-1984-15188-7 |mr=722856 }}

[[Category:Algebra]]
[[Category:Invariant theory]]