Invariant theory bcFrom Wikipedia, the free encyclopedia

Contents

1 Bracket algebra 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Bracket ring 22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Canonizant 33.1 Canonizants of a binary form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Capelli’s identity 44.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Relations with representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.2.1 Case m = 1 and representation Sk Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2.2 The universal enveloping algebra U(gln) and its center . . . . . . . . . . . . . . . . . . . 64.2.3 General m and dual pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.1 Turnbull’s identity for symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.2 The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices . . . . . . . . . . . . 104.3.3 The Caracciolo–Sportiello–Sokal identity for Manin matrices . . . . . . . . . . . . . . . . 104.3.4 The Mukhin–Tarasov–Varchenko identity and the Gaudin model . . . . . . . . . . . . . . 104.3.5 Permanents, immanants, traces – “higher Capelli identities” . . . . . . . . . . . . . . . . . 11

4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Catalecticant 145.1 Binary forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Catalecticants of quartic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Cayley’s Ω process 16

i

ii CONTENTS

6.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Chevalley–Iwahori–Nagata theorem 187.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Chevalley–Shephard–Todd theorem 198.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Invariant theory 219.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 The nineteenth-century origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.3 Hilbert’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.4 Geometric invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 The Classical Groups 2510.1 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 27

10.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 1

Bracket algebra

In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra witha symbolic representation of projective invariants.Given that L is a proper signed alphabet and Super[L] is the supersymmetric algebra, the bracket algebra Bracket[L]of dimension n over the field K is the quotient of the algebra BraceL obtained by imposing the congruence relationsbelow, where w, w', ..., w" are any monomials in Super[L]:

1. w = 0 if length(w) ≠ n

2. ww'...w" = 0whenever any positive letter a ofL occursmore than n times in themonomial ww'...w".

3. Let ww'...w" be a monomial in BraceL in which some positive letter a occurs more than n times, andlet b, c, d, e, ..., f, g be any letters in L.

1.1 See also• Bracket ring

1.2 References• Anick, David; Rota, Gian-Carlo (September 15, 1991), “Higher-Order Syzygies for the Bracket Algebra andfor the Ring of Coordinates of the Grassmanian”, Proceedings of the National Academy of Sciences 88 (18):8087–8090, doi:10.1073/pnas.88.18.8087, ISSN 0027-8424, JSTOR 2357546.

• Huang, Rosa Q.; Rota, Gian-Carlo; Stein, Joel A. (1990), “Supersymmetric Bracket Algebra and Invariant The-ory”,Acta ApplicandaeMathematicae (Kluwer Academic Publishers) 21: 193–246, doi:10.1007/BF00053298.

1

Chapter 2

Bracket ring

In mathematics, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d by dminors of a generic d by n matrix (xij).The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plückerembedding.[1]

For given d ≤ n we define as formal variables the brackets [λ1 λ2 ... λd] with the λ taken from 1,...,n, subject to [λ1λ2 ... λd] = − [λ2 λ1 ... λd] and similarly for other transpositions. The set Λ(n,d) of size

(nd

)generates a polynomial

ring K[Λ(n,d)] over a field K. There is a homomorphism Φ(n,d) from K[Λ(n,d)] to the polynomial ring K[xi,j] innd indeterminates given by mapping [λ1 λ2 ... λd] to the determinant of the d by d matrix consisting of the columnsof the xi,j indexed by the λ. The bracket ring B(n,d) is the image of Φ. The kernel I(n,d) of Φ encodes the relationsor syzygies that exist between the minors of a generic n by d matrix. The projective variety defined by the ideal I isthe (n−d)d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensionalspace.[2]

To compute with brackets it is necessary to determine when an expression lies in the ideal I(n,d). This is achievedby a straightening law due to Young (1928).[3]

2.1 See also• Bracket algebra

2.2 References[1] Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Ency-

clopedia of Mathematics and Its Applications 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl0944.52006

[2] Sturmfels (2008) pp.78–79

[3] Sturmfels (2008) p.80

• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525, Zbl 0196.05802

• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102, Zbl 0258.14011

• Sturmfels, Bernd (2008), Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation(2nd ed.), Springer-Verlag, ISBN 3211774165, Zbl 1154.13003

• Sturmfels, Bernd; White, Neil (1990), “Stanley decompositions of the bracket ring”,Mathematica Scandinavica67 (2): 183–189, ISSN 0025-5521, MR 1096453, Zbl 0727.13005

2

Chapter 3

Canonizant

In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical formfor them.

3.1 Canonizants of a binary form

The canonizant of a binary form of degree 2n – 1 is a covariant of degree n and order n, given by the catalecticant ofthe penultimate emanant, which is the determinant of the n by n Hadamard matrix with entries ai₊jx + ai₊j₊₁y for 0≤ i,j < n.

3.2 References• Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. ClarendonPress, JFM 26.0135.01

3

Chapter 4

Capelli’s identity

In mathematics, Capelli’s identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) =det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebragln . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley’s Ω process.

4.1 Statement

Suppose that xij for i,j = 1,...,n are commuting variables. Write E ᵢ for the polarization operator

Eij =

n∑a=1

xia∂

∂xja.

The Capelli identity states that the following differential operators, expressed as determinants, are equal:

∣∣∣∣∣∣∣∣∣E11 + n− 1 · · · E1,n−1 E1n

... . . . ......

En−1,1 · · · En−1,n−1 + 1 En−1,n

En1 · · · En,n−1 Enn + 0

∣∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣x11 · · · x1n... . . . ...xn1 · · · xnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂∂x11

· · · ∂∂x1n... . . . ...

∂∂xn1

· · · ∂∂xnn

∣∣∣∣∣∣∣.Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded withall terms preserving their “left to right” order. Such a determinant is often called a column-determinant, since it canbe obtained by the column expansion of the determinant starting from the first column. It can be formally written as

det(A) =∑σ∈Sn

sgn(σ)Aσ(1),1Aσ(2),2 · · ·Aσ(n),n,

where in the product first come the elements from the first column, then from the second and so on. The determinanton the far right is Cayley’s omega process, and the one on the left is the Capelli determinant.The operators E ᵢ can be written in a matrix form:

E = XDt,

where E,X,D are matrices with elements E ᵢ , xᵢ , ∂∂xij

respectively. If all elements in these matrices would becommutative then clearly det(E) = det(X) det(Dt) . The Capelli identity shows that despite noncommutativitythere exists a “quantization” of the formula above. The only price for the noncommutivity is a small correction:(n− i)δij on the left hand side. For generic noncommutative matrices formulas like

4

4.2. RELATIONS WITH REPRESENTATION THEORY 5

det(AB) = det(A) det(B)

do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices.That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof doesnot seem to exist. Direct verification of the statement can be given as an exercise for n' = 2, but is already long for n= 3.

4.2 Relations with representation theory

Consider the following slightly more general context. Suppose that n and m are two integers and xij for i =1, . . . , n, j = 1, . . . ,m , be commuting variables. Redefine Eij by almost the same formula:

Eij =

m∑a=1

xia∂

∂xja.

with the only difference that summation index a ranges from 1 tom . One can easily see that such operators satisfythe commutation relations:

[Eij , Ekl] = δjkEil − δilEkj .

Here [a, b] denotes the commutator ab − ba . These are the same commutation relations which are satisfied by thematrices eij which have zeros everywhere except the position (i, j) , where 1 stands. ( eij are sometimes calledmatrix units). Hence we conclude that the correspondence π : eij 7→ Eij defines a representation of the Lie algebragln in the vector space of polynomials of xij .

4.2.1 Case m = 1 and representation Sk Cn

It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:

Eij = xi∂

∂xj.

In particular, for the polynomials of the first degree it is seen that:

Eijxk = δjkxi.

Hence the action of Eij restricted to the space of first-order polynomials is exactly the same as the action of matrixunits eij on vectors in Cn . So, from the representation theory point of view, the subspace of polynomials of firstdegree is a subrepresentation of the Lie algebra gln , which we identified with the standard representation in Cn .Going further, it is seen that the differential operators Eij preserve the degree of the polynomials, and hence thepolynomials of each fixed degree form a subrepresentation of the Lie algebra gln . One can see further that the spaceof homogeneous polynomials of degree k can be identified with the symmetric tensor power SkCn of the standardrepresentation Cn .One can also easily identify the highest weight structure of these representations. The monomial xk1 is a highest weightvector, indeed: Eijxk1 = 0 for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: Eiixk1 = kδi1x

k1 .

Such representation is sometimes called bosonic representation of gln . Similar formulas Eij = ψi∂∂ψj

define theso-called fermionic representation, here ψi are anti-commuting variables. Again polynomials of k-th degree form anirreducible subrepresentation which is isomorphic to ΛkCn i.e. anti-symmetric tensor power of Cn . Highest weightof such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representationsof gln .

6 CHAPTER 4. CAPELLI’S IDENTITY

Capelli identity for m = 1

Let us return to the Capelli identity. One can prove the following:

det(E + (n− i)δij) = 0, n > 1

the motivation for this equality is the following: consider Ecij = xipj for some commuting variables xi, pj . Thematrix Ec is of rank one and hence its determinant is equal to zero. Elements of matrix E are defined by thesimilar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity:det(Ec) = 0 can be preserved for the small price of correcting matrix E by (n− i)δij .Let us also mention that similar identity can be given for the characteristic polynomial:

det(t+ E + (n− i)δij) = t[n] + Tr(E)t[n−1],

where t[k] = t(t+1) · · · (t+k−1) . The commutative counterpart of this is a simple fact that for rank = 1 matricesthe characteristic polynomial contains only the first and the second coefficients.Let us consider an example for n = 2.

∣∣∣∣t+ E11 + 1 E12

E21 t+ E22

∣∣∣∣ = ∣∣∣∣t+ x1∂1 + 1 x1∂2x2∂1 t+ x2∂2

∣∣∣∣= (t+ x1∂1 + 1)(t+ x2∂2)− x2∂1x1∂2

= t(t+ 1) + t(x1∂1 + x2∂2) + x1∂1x2∂2 + x2∂2 − x2∂1x1∂2

Using

∂1x1 = x1∂1 + 1, ∂1x2 = x2∂1, x1x2 = x2x1

we see that this is equal to:

t(t+ 1) + t(x1∂1 + x2∂2) + x2x1∂1∂2 + x2∂2 − x2x1∂1∂2 − x2∂2

= t(t+ 1) + t(x1∂1 + x2∂2) = t[2] + tTr(E).

4.2.2 The universal enveloping algebra U(gln) and its center

An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is the commutator[Eij , det(E + (n− i)δij)] = 0 is equal to zero. It can be generalized:Consider any elements Eij in any ring, such that they satisfy the commutation relation [Eij , Ekl] = δjkEil − δilEkj, (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows:

det(t+ E + (n− i)δij) = t[n] +∑

k=n−1,...,0

t[k]Ck,

where t[k] = t(t+ 1) · · · (t+ k − 1),

then:

• elements Ck commute with all elements Eij

• elements Ck can be given by the formulas similar to the commutative case:

4.2. RELATIONS WITH REPRESENTATION THEORY 7

Ck =∑

I=(i1<i2<···<ik)

det(E + (k − i)δij)II ,

i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction +(k − i)δij . In particularelement C0 is the Capelli determinant considered above.These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the directfew lines short proof does not seem to exist, despite the simplicity of the formulation.The universal enveloping algebra

U(gln)

can defined as an algebra generated by

Eij

subject to the relations

[Eij , Ekl] = δjkEil − δilEkj

alone. The proposition above shows that elements Ckbelong to the center of U(gln) . It can be shown that theyactually are free generators of the center of U(gln) . They are sometimes called Capelli generators. The Capelliidentities for them will be discussed below.Consider an example for n = 2.

∣∣∣∣t+ E11 + 1 E12

E21 t+ E22

∣∣∣∣ = (t+ E11 + 1)(t+ E22)− E21E12

= t(t+ 1) + t(E11 + E22) + E11E22 − E21E12 + E22.

It is immediate to check that element (E11 + E22) commute with Eij . (It corresponds to an obvious fact that theidentity matrix commute with all other matrices). More instructive is to check commutativity of the second elementwith Eij . Let us do it for E12 :

[E12, E11E22 − E21E12 + E22]

= [E12, E11]E22 + E11[E12, E22]− [E12, E21]E12 − E21[E12, E12] + [E12, E22]

= −E12E22 + E11E12 − (E11 − E22)E12 − 0 + E12

= −E12E22 + E22E12 + E12 = −E12 + E12 = 0.

We see that the naive determinant E11E22 −E21E12 will not commute with E12 and the Capelli’s correction+E22

is essential to ensure the centrality.

4.2.3 General m and dual pairs

Let us return to the general case:

Eij =m∑a=1

xia∂

∂xja,

for arbitrary n and m. Definition of operators E ᵢ can be written in a matrix form: E = XDt , where E is n × nmatrix with elements Eij ; X is n×m matrix with elements xij ; D is n×m matrix with elements ∂

∂xij.

8 CHAPTER 4. CAPELLI’S IDENTITY

Capelli–Cauchy–Binet identitiesFor general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements ofthese matrices would commute then one knows that the determinant of E can be expressed by the so-called Cauchy–Binet formula via minors of X and D. An analogue of this formula also exists for matrix E again for the same mildprice of the correction E → (E + (n− i)δij) :

det(E + (n− i)δij) =∑

I=(1≤i1<i2<···<in≤m)

det(XI) det(DtI)

In particular (similar to the commutative case): if m<n'', then <math>\det(E+(n-i)\delta_ij) =0 </math>; if m=nwe return to the identity above.

Let us also mention that similar to the commutative case (see Cauchy–Binet for minors), one can express not onlythe determinant of E, but also its minors via minors of X and D:

det(E + (s− i)δij)KL =∑

I=(1≤i1<i2<···<is≤m)

det(XKI) det(DtIL)

Here K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usuallyMKL denotes a submatrixof M formed by the elements Mkalb. Pay attention that the Capelli correction now contains s, not n as in previousformula. Note that for s=1, the correction (s − i) disappears and we get just the definition of E as a product of X andtranspose to D. Let us also mention that for generic K,L corresponding minors do not commute with all elements Eij,so the Capelli identity exists not only for central elements.As a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention thefollowing:

det(t+ E + (n− i)δij) = t[n] +∑

k=n−1,...,0

t[k]∑I,J

det(XIJ) det(DtJI),

where I = (1 ≤ i1 < · · · < ik ≤ n), J = (1 ≤ j1 < · · · < jk ≤ n) . This formula is similar to the commutativecase, modula +(n− i)δij at the left hand side and t[n] instead of tn at the right hand side.Relation to dual pairsModern interest in these identities has been much stimulated by Roger Howe who considered them in his theoryof reductive dual pairs (also known as Howe duality). To make the first contact with these ideas, let us look moreprecisely on operators Eij . Such operators preserve the degree of polynomials. Let us look at the polynomials ofdegree 1: Eijxkl = xilδjk , we see that index l is preserved. One can see that from the representation theory pointof view polynomials of the first degree can be identified with direct sum of the representations Cn ⊕ · · · ⊕Cn , herel-th subspace (l=1...m) is spanned by xil , i = 1, ..., n. Let us give another look on this vector space:

Cn ⊕ · · · ⊕ Cn = Cn ⊗ Cm.

Such point of view gives the first hint of symmetry between m and n. To deepen this idea let us consider:

Edualij =

n∑a=1

xai∂

∂xaj.

These operators are given by the same formulas asEij modula renumeration i↔ j , hence by the same arguments wecan deduce that Edual

ij form a representation of the Lie algebra glm in the vector space of polynomials of xij. Beforegoing further we can mention the following property: differential operatorsEdual

ij commute with differential operatorsEkl .The Lie groupGLn×GLm acts on the vector spaceCn⊗Cm in a natural way. One can show that the correspondingaction of Lie algebra gln × glm is given by the differential operators Eij and Edual

ij respectively. This explains thecommutativity of these operators.The following deeper properties actually hold true:

4.3. GENERALIZATIONS 9

• The only differential operators which commute with Eij are polynomials in Edualij , and vice versa.

• Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible repre-sentations of GLn and GLm can be given as follows:

C[xij ] = S(Cn ⊗ Cm) =∑D

ρDn ⊗ ρD′

m .

The summands are indexed by the Young diagrams D, and representations ρD are mutually non-isomorphic. Anddiagram D determine D′ and vice versa.

• In particular the representation of the big group GLn × GLm is multiplicity free, that is each irreduciblerepresentation occurs only one time.

One easily observe the strong similarity to Schur–Weyl duality.

4.3 Generalizations

Much work have been done on the identity and its generalizations. Approximately two dozens of mathematicians andphysicists contributed to the subject, to name a few: R. Howe, B. Kostant[1][2] Fields medalist A. Okounkov[3][4] A.Sokal,[5] D. Zeilberger.[6]

It seems historically the first generalizations were obtained by Herbert Westren Turnbull in 1948,[7] who found thegeneralization for the case of symmetric matrices (see[5][6] for modern treatments).The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view.Such generalizations consist of changing Lie algebra gln to simple Lie algebras [8] and their super[9][10] (q),[11][12] andcurrent versions.[13] As well as identity can be generalized for different reductive dual pairs.[14][15] And finally one canconsider not only the determinant of the matrix E, but its permanent,[16] trace of its powers and immanants.[3][4][17][18]Let us mention few more papers;[19][20][21] [22] [23] [24] [25] still the list of references is incomplete. It has been believedfor quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purelyalgebraic generalization of the identity have been found in 2008[5] by S. Caracciolo, A. Sportiello, A. D. Sokal whichhas nothing to do with any Lie algebras.

4.3.1 Turnbull’s identity for symmetric matrices

Consider symmetric matrices

X =

∣∣∣∣∣∣∣∣∣∣∣

x11 x12 x13 · · · x1nx12 x22 x23 · · · x2nx13 x23 x33 · · · x3n...

...... . . . ...

x1n x2n x3n · · · xnn

∣∣∣∣∣∣∣∣∣∣∣, D =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

2 ∂∂x11

∂∂x12

∂∂x13

· · · ∂∂x1n

∂∂x12

2 ∂∂x22

∂∂x23

· · · ∂∂x2n

∂∂x13

∂∂x23

2 ∂∂x33

· · · ∂∂x3n

......

... . . . ...∂

∂x1n

∂∂x2n

∂∂x3n

· · · 2 ∂∂xnn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣Herbert Westren Turnbull[7] in 1948 discovered the following identity:

det(XD + (n− i)δij) = det(X) det(D)

Combinatorial proof can be found in the paper,[6] another proof and amusing generalizations in the paper,[5] see alsodiscussion below.

10 CHAPTER 4. CAPELLI’S IDENTITY

4.3.2 The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices

Consider antisymmetric matrices

X =

∣∣∣∣∣∣∣∣∣∣∣

0 x12 x13 · · · x1n−x12 0 x23 · · · x2n−x13 −x23 0 · · · x3n...

...... . . . ...

−x1n −x2n −x3n · · · 0

∣∣∣∣∣∣∣∣∣∣∣, D =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

0 ∂∂x12

∂∂x13

· · · ∂∂x1n

− ∂∂x12

0 ∂∂x23

· · · ∂∂x2n

− ∂∂x13

− ∂∂x23

0 · · · ∂∂x3n

......

... . . . ...− ∂∂x1n

− ∂∂x2n

− ∂∂x3n

· · · 0

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣.

Then

det(XD + (n− i)δij) = det(X) det(D).

4.3.3 The Caracciolo–Sportiello–Sokal identity for Manin matrices

Consider two matrices M and Y over some associative ring which satisfy the following condition

[Mij , Ykl] = −δjkQil

for some elements Qil. Or ”in words”: elements in j-th column ofM commute with elements in k-th row of Y unlessj = k, and in this case commutator of the elements Mik and Ykl depends only on i, l, but does not depend on k.Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements).Then for the square matrix case

det(MY +Q diag(n− 1, n− 2, . . . , 1, 0)) = det(M) det(Y ).

Here Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n− 1, n − 2, ..., 1, 0 on the diagonal.See [5] proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.Obviously the original Cappeli’s identity the particular case of this identity. Moreover from this identity one can seethat in the original Capelli’s identity one can consider elements

∂

∂xij+ fij(x11, . . . , xkl, . . . )

for arbitrary functions fij and the identity still will be true.

4.3.4 The Mukhin–Tarasov–Varchenko identity and the Gaudin model

Statement

Consider matrices X and D as in Capelli’s identity, i.e. with elements xij and ∂ij at position (ij).Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complexnumbers.

det(∂

∂z−A−X

1

z −BDt

)

4.3. GENERALIZATIONS 11

= detcommute all if as calculateall Putx and zright the on derivations all while left, the on(∂

∂z−A−X

1

z −BDt

)Here the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries.The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, whilederivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics).

The Gaudin quantum integrable system and Talalaev’s theorem

The matrix

L(z) = A+X1

z −BDt

is a Lax matrix for the Gaudin quantum integrable spin chain system. D. Talalaev solved the long-standing problemof the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model, discoveringthe following theorem.Consider

det(∂

∂z− L(z)

)=

n∑i=0

Hi(z)

(∂

∂z

)i.

Then for all i,j,z,w

[Hi(z),Hj(w)] = 0,

i.e. Hi(z) are generating functions in z for the differential operators in x which all commute. So they provide quantumcommuting conservation laws for the Gaudin model.

4.3.5 Permanents, immanants, traces – “higher Capelli identities”

The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanents,immanants and traces. Based on the combinatorial approach paper by S.G. Williamson [26] was one of the first resultsin this direction.

Turnbull’s identity for permanents of antisymmetric matrices

Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of theHUKS identity above.Then

perm(XtD − (n− i)δij) = permcommute all if as Calculateall Putxright the on derivations all with left, the on (X

tD).

Let us cite:[6] "...is stated without proof at the end of Turnbull’s paper”. The authors themselves follow Turnbull – atthe very end of their paper they write:“Since the proof of this last identity is very similar to the proof of Turnbull’s symmetric analog (with a slight twist),we leave it as an instructive and pleasant exercise for the reader.”.The identity is deeply analyzed in paper .[27]

12 CHAPTER 4. CAPELLI’S IDENTITY

4.4 References[1] Kostant, B.; Sahi, S. (1991), “The Capelli Identity, tube domains, and the generalized Laplace transform”, Advances in

Math. 87: 71–92, doi:10.1016/0001-8708(91)90062-C

[2] Kostant, B.; Sahi, S. (1993), “Jordan algebras andCapelli identities”, InventionesMathematicae 112 (1): 71–92, doi:10.1007/BF01232451

[3] Okounkov, A. (1996), Quantum Immanants and Higher Capelli Identities, arXiv:q-alg/9602028

[4] Okounkov, A. (1996), Young Basis, Wick Formula, and Higher Capelli Identities, arXiv:q-alg/9602027

[5] Caracciolo, S.; Sportiello, A.; Sokal, A. (2008), Noncommutative determinants, Cauchy–Binet formulae, and Capelli-typeidentities. I. Generalizations of the Capelli and Turnbull identities, arXiv:0809.3516

[6] Foata, D.; Zeilberger, D. (1993), Combinatorial Proofs of Capelli’s and Turnbull’s Identities from Classical Invariant Theory,arXiv:math/9309212

[7] Turnbull, HerbertWestren (1948), “Symmetric determinants and the Cayley and Capelli operators”, Proc. EdinburghMath.Soc. 8 (2): 76–86, doi:10.1017/S0013091500024822

[8] Molev, A.; Nazarov, M. (1997), Capelli Identities for Classical Lie Algebras, arXiv:q-alg/9712021

[9] Molev, A. (1996), Factorial supersymmetric Schur functions and super Capelli identities, arXiv:q-alg/9606008

[10] Nazarov, M. (1996), Capelli identities for Lie superalgebras, arXiv:q-alg/9610032

[11] Noumi, M.; Umeda, T.; Wakayma, M. (1994), “A quantum analogue of the Capelli identity and an elementary differentialcalculus on GLq(n)", Duke Mathematical Journal 76 (2): 567–594, doi:10.1215/S0012-7094-94-07620-5

[12] Noumi, M.; Umeda, T.; Wakayma, M. (1996), “Dual pairs, spherical harmonics and a Capelli identity in quantum grouptheory”, Compositio Mathematica 104 (2): 227–277

[13] Mukhin, E.; Tarasov, V.; Varchenko, A. (2006), A generalization of the Capelli identity, arXiv:math.QA/0610799

[14] Itoh,M. (2004), “Capelli identities for reductive dual pairs”,Advances inMathematics 194 (2): 345–397, doi:10.1016/j.aim.2004.06.010

[15] Itoh, M. (2005), “Capelli Identities for the dual pair ( O M, Sp N)", Mathematische Zeitschrift 246 (1–2): 125–154,doi:10.1007/s00209-003-0591-2

[16] Nazarov, M. (1991), “Quantum Berezinian and the classical Capelli identity”, Letters in Mathematical Physics 21 (2): 123–131, doi:10.1007/BF00401646

[17] Nazarov, M. (1996), Yangians and Capelli identities, arXiv:q-alg/9601027

[18] Molev, A. (1996), A Remark on the Higher Capelli Identities, arXiv:q-alg/9603007

[19] Kinoshita, K.; Wakayama, M. (2002), “Explicit Capelli identities for skew symmetric matrices”, Proceedings of the Edin-burgh Mathematical Society 45 (2): 449–465, doi:10.1017/S0013091500001176

[20] Hashimoto, T. (2008),Generating function for GLn-invariant differential operators in the skewCapelli identity, arXiv:0803.1339

[21] Nishiyama, K.; Wachi, A. (2008), A note on the Capelli identities for symmetric pairs of Hermitian type, arXiv:0808.0607

[22] Umeda, Toru (2008), “On the proof of the Capelli identities”, Funkcialaj Ekvacioj 51 (1): 1–15, doi:10.1619/fesi.51.1

[23] Brini, A; Teolis, A (1993), “Capelli’s theory, Koszulmaps, and superalgebras”, PNAS 90 (21): 10245–10249, doi:10.1073/pnas.90.21.10245

[24] Koszul, J (1981), “Les algebres de Lie graduées de type sl (n, 1) et l'opérateur de A. Capelli”, C.R. Acad. Sci. Paris (292):139–141

[25] Orsted, B; Zhang, G (2001), Capelli identity and relative discrete series of line bundles over tube domains

[26] Williamson, S. (1981), “Symmetry operators, polarizations, and a generalized Capelli identity”, Linear & Multilinear Al-gebra 10 (2): 93–102, doi:10.1080/03081088108817399

[27] Umeda, Toru (2000), “On Turnbull identity for skew-symmetric matrices”, Proc. Edinburgh Math. Soc. 43 (2): 379–393,doi:10.1017/S0013091500020988

4.5. FURTHER READING 13

4.5 Further reading• Capelli, Alfredo (1887), “Ueber die Zurückführung der Cayley’schen Operation Ω auf gewöhnliche Polar-Operationen”,Mathematische Annalen (Berlin / Heidelberg: Springer) 29 (3): 331–338, doi:10.1007/BF01447728,ISSN 1432-1807

• Howe, Roger (1989), “Remarks on classical invariant theory”, Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR2001418, MR 0986027

• Howe, Roger; Umeda, Toru (1991), “The Capelli identity, the double commutant theorem, and multiplicity-free actions”, Mathematische Annalen 290 (1): 565–619, doi:10.1007/BF01459261

• Umeda, Tôru (1998), “The Capelli identities, a century after”, Selected papers on harmonic analysis, groups,and invariants, Amer. Math. Soc. Transl. Ser. 2 183, Providence, R.I.: Amer. Math. Soc., pp. 51–78, ISBN978-0-8218-0840-5, MR 1615137

• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255, retrieved 03/2007/26 Check date values in: |accessdate=(help)

Chapter 5

Catalecticant

But the catalecticant of the biquadratic function of x, y was first brought into notice as an invariant by Mr Boole;and the discriminant of the quadratic function of x, y is identical with its catalecticant, as also with its Hessian.Meicatalecticizant would more completely express the meaning of that which, for the sake of brevity, I denominatethe catalecticant.Sylvester (1852), quoted by (Miller 2010)

In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coefficients thatvanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced bySylvester (1852); see (Miller 2010). The word catalectic refers to an incomplete line of verse, lacking a syllable atthe end or ending with an incomplete foot.

5.1 Binary forms

The catalecticant of a binary form of degree 2n is a polynomial in its coefficients that vanishes when the binary formis a sum of at most n powers of linear forms Sturmfels (1993).The catalecticant of a binary form can be given as the determinant of a catalecticant matrix (Eisenbud 1988), alsocalled a Hankel matrix, that is a square matrix with constant (positive sloping) skew-diagonals, such as

a b c d eb c d e fc d e f gd e f g he f g h i

.

5.2 Catalecticants of quartic forms

The catalecticant of a quartic form is the resultant of its second partial derivatives. For binary quartics the catalecticantvanishes when the form is a sum of 2 4th powers. For a ternary quartic the catalecticant vanishes when the form is asum of 5 4th powers. For quaternary quartics the catalecticant vanishes when the form is a sum of 9 4th powers. Forquinary quartics the catalecticant vanishes when the form is a sum of 14 4th powers. (Elliot 1915, p.295)

5.3 References

• Eisenbud, David (1988), “Linear sections of determinantal varieties”, American Journal of Mathematics 110(3): 541–575, doi:10.2307/2374622, ISSN 0002-9327, MR 944327

14

5.4. EXTERNAL LINKS 15

• Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. ClarendonPress, JFM 26.0135.01

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN978-3-211-82445-0,MR1255980

• Sylvester, J. J. (1852), “On the principles of the calculus of forms”, Cambridge and Dublin MathematicalJournal: 52–97

5.4 External links• Miller, Jeff (2010), Earliest Known Uses of Some of the Words of Mathematics (C)

Chapter 6

Cayley’s Ω process

This article is about the mathematical process. For the industrial OMEGA process, see OMEGA process.

Inmathematics,Cayley’sΩ process, introduced byArthur Cayley (1846), is a relatively invariant differential operatoron the general linear group, that is used to construct invariants of a group action.As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by the determinant

Ω =

∣∣∣∣∣∣∣∂

∂x11· · · ∂

∂x1n... . . . ...∂

∂xn1· · · ∂

∂xnn

∣∣∣∣∣∣∣.For binary forms f in x1, y1 and g in x2, y2 the Ω operator is ∂2fg

∂x1∂y2− ∂2fg

∂x2∂y1. The r-fold Ω process Ωr(f, g) on

two forms f and g in the variables x and y is then

1. Convert f to a form in x1, y1 and g to a form in x2, y2

2. Apply the Ω operator r times to the function fg, that is, f times g in these four variables

3. Substitute x for x1 and x2, y for y1 and y2 in the result

The result of the r-fold Ω process Ωr(f, g) on the two forms f and g is also called the r-th transvectant and is commonlywritten (f, g)r.

6.1 Applications

Cayley’s Ω process appears in Capelli’s identity, which Weyl (1946) used to find generators for the invariants ofvarious classical groups acting on natural polynomial algebras.Hilbert (1890) used Cayley’s Ω process in his proof of finite generation of rings of invariants of the general lineargroup. His use of the Ω process gives an explicit formula for the Reynolds operator of the special linear group.Cayley’s Ω process is used to define transvectants.

6.2 References

• Cayley, Arthur (1846), “On linear transformations”, Cambridge and Dublin mathematical journal 1: 104–122Reprinted in Cayley (1889), The collected mathematical papers 1, Cambridge: Cambridge University press,pp. 95–112

16

6.2. REFERENCES 17

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

• Howe, Roger (1989), “Remarks on classical invariant theory.”, Transactions of the American Mathematical So-ciety (American Mathematical Society) 313 (2): 539–570, doi:10.1090/S0002-9947-1989-0986027-X, ISSN0002-9947, JSTOR 2001418, MR 0986027

• Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1

• Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, New York: Springer-Verlag, ISBN 978-3-211-82445-0, MR 1255980

• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255, retrieved 03/2007/26 Check date values in: |accessdate=(help)

Chapter 7

Chevalley–Iwahori–Nagata theorem

In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearlyon a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomialsis an isomorphism if this ring is finitely generated and all orbits of G on V are closed (Dieudonné & Carrell 1970,1971, p.55). It is named after Claude Chevalley, Nagayoshi Iwahori, and Masayoshi Nagata.

7.1 References• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525

• Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

18

Chapter 8

Chevalley–Shephard–Todd theorem

In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring ofinvariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generatedby pseudoreflections. In the case of subgroups of the complex general linear group the theorem was first proved byG. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwardsgave a uniform proof. It has been extended to finite linear groups over an arbitrary field in the non-modular case byJean-Pierre Serre.

8.1 Statement of the theorem

Let V be a finite-dimensional vector space over a field K and let G be a finite subgroup of the general linear groupGL(V). An element s of GL(V) is called a pseudoreflection if it fixes a codimension 1 subspace of V and is not theidentity transformation I, or equivalently, if the kernel Ker (s − I) has codimension one in V. Assume that the orderof G is relatively prime to the characteristic of K (the so-called non-modular case). Then the following properties areequivalent:[1]

• (A) The group G is generated by pseudoreflections.

• (B) The algebra of invariants K[V]G is a (free) polynomial algebra.

• (B′) The algebra of invariants K[V]G is a regular ring.

• (C) The algebra K[V] is a free module over K[V]G.

• (C′) The algebra K[V] is a projective module over K[V]G.

In the case when the field K is the field C of complex numbers, the first condition is usually stated as "G is a complexreflection group". Shephard and Todd derived a full classification of such groups.

8.2 Examples• Let V be one-dimensional. Then any finite group faithfully acting on V is a subgroup of the multiplicativegroup of the field K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n,where n is its order, so G is generated by pseudoreflections. In this case, K[V] = K[x] is the polynomial ringin one variable and the algebra of invariants of G is the subalgebra generated by xn, hence it is a polynomialalgebra.

• LetV =Kn be the standard n-dimensional vector space andG be the symmetric group Sn acting by permutationsof the elements of the standard basis. The symmetric group is generated by transpositions (ij), which act byreflections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants isthe polynomial algebra generated by the elementary symmetric functions e1, … en.

19

20 CHAPTER 8. CHEVALLEY–SHEPHARD–TODD THEOREM

• Let V = K2 and G be the cyclic group of order 2 acting by ±I. In this case, G is not generated by pseudoreflec-tions, since the nonidentity element s of G acts without fixed points, so that dim Ker (s − I) = 0. On the otherhand, the algebra of invariants is the subalgebra of K[V] = K[x, y] generated by the homogeneous elementsx2, xy, and y2 of degree 2. This subalgebra is not a polynomial algebra because of the relation x2y2 = (xy)2.

8.3 Generalizations

Broer (2007) gave an extension of the Chevalley–Shephard–Todd theorem to positive characteristic.There has been much work on the question of when a reductive algebraic group acting on a vector space has apolynomial ring of invariants. In the case when the algebraic group is simple all cases when the invariant ring ispolynomial have been classified by Schwarz (1978)In general, the ring of invariants of a finite group acting linearly on a complex vector space is Cohen-Macaulay, so itis a finite rank free module over a polynomial subring.

8.4 Notes[1] See, e.g.: Bourbaki, Lie, chap. V, §5, nº5, theorem 4 for equivalence of (A), (B) and (C); page 26 of for equivalence of

(A) and (B′); pages 6–18 of for equivalence of (C) and (C′) for a proof of (B′)⇒(A).

8.5 References• Bourbaki, Nicolas, Éléments de mathématiques : Groupes et algèbres de Lie (English translation: Bourbaki,Nicolas, Elements of Mathematics: Lie Groups and Lie Algebras)

• Broer, Abraham (2007), On Chevalley-Shephard-Todd’s theorem in positive characteristic, [], arXiv:0709.0715

• Chevalley, Claude (1955), “Invariants of finite groups generated by reflections”, Amer. J. Of Math. 77 (4):778–782, doi:10.2307/2372597, JSTOR 2372597

• Neusel, Mara D.; Smith, Larry (2002), Invariant Theory of Finite Groups, American Mathematical Society,ISBN 0-8218-2916-5

• Shephard, G. C.; Todd, J. A. (1954), “Finite unitary reflection groups”, Canadian J. Math. 6: 274–304,doi:10.4153/CJM-1954-028-3

• Schwarz, G. (1978), “Representations of simple Lie groups with regular rings of invariants”, Invent. Math. 49(2): 167–191, doi:10.1007/BF01403085

• Smith, Larry (1997), “Polynomial invariants of finite groups. A survey of recent developments”, Bull. Amer.Math. Soc. 34 (3): 211–250, doi:10.1090/S0273-0979-97-00724-6, MR 1433171

• Springer, T. A. (1977), Invariant Theory, Springer, ISBN 0-387-08242-5

Chapter 9

Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vectorspaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicitdescription of polynomial functions that do not change, or are invariant, under the transformations from a givenlinear group. For example, if we consider the action of the special linear group SLn on the space of n by n matricesby left multiplication, then the determinant is an invariant of this action because the determinant of A X equals thedeterminant of X, when A is in SLn.

9.1 Introduction

LetG be a group, andV a finite-dimensional vector space over a field k (which in classical invariant theory was usuallyassumed to be the complex numbers). A representation of G in V is a group homomorphism π : G → GL(V ) ,which induces a group action of G on V. If k[V] is the space of polynomial functions on V, then the group action ofG on V produces an action on k[V] by the following formula:

(g · f)(x) := f(g−1x) ∀x ∈ V, g ∈ G, f ∈ k[V ].

With this action it is natural to consider the subspace of all polynomial functions which are invariant under this groupaction, in other words the set of polynomials such that g.f = f for all g in G. This space of invariant polynomials isdenoted k[V]G.First problem of invariant theory:[1] Is k[V]G a finitely generated algebra over k?For example, ifG=SLn andV=Mn to space of square matrices, and the action ofG onV is given by left multiplication,then k[V]G is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, inthis case, every invariant polynomial is a linear combination of power of the determinant polynomial. So in this case,k[V]G is finitely generated over k.If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomialrelations between the basis elements (known as the syzygies) is finitely generated over k[V].Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results wasthe main theorem on the symmetric functions that described the invariants of the symmetric group Sn acting onthe polynomial ring R[x1, …, xn] by permutations of the variables. More generally, the Chevalley–Shephard–Toddtheorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invarianttheory of finite groups emphasizes “effective” results, such as explicit bounds on the degrees of the generators. Thecase of positive characteristic, ideologically close to modular representation theory, is an area of active study, withlinks to algebraic topology.Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theoriesof quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, whereinvariant theory was expected to play a major role in organizing the material. One of the highlights of this relationshipis the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory.

21

22 CHAPTER 9. INVARIANT THEORY

David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creationof a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the same questions inmore constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideasback to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory. In largemeasure due to the influence of Mumford, the subject of invariant theory is seen to encompass the theory of actionsof linear algebraic groups on affine and projective varieties. A distinct strand of invariant theory, going back to theclassical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rotaand his school. A prominent example of this circle of ideas is given by the theory of standard monomials.

9.2 The nineteenth-century origins

The theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: agrown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley’s Jovian head.Weyl (1939b, p.489)

Cayley, whose fundamental work establishing “invariant theory” was “On the Theory of Linear Transformations(1845).” In the opening of his paper, Cayley credits an 1841 paper of George Boole, “investigations were suggestedto me by a very elegant paper on the same subject... byMr Boole.” (Boole’s paper was Exposition of a General Theoryof Linear Transformations, Cambridge Mathematical Journal.)Classically, the term “invariant theory” refers to the study of invariant algebraic forms (equivalently, symmetric ten-sors) for the action of linear transformations. This was a major field of study in the latter part of the nineteenthcentury. Current theories relating to the symmetric group and symmetric functions, commutative algebra, modulispaces and the representations of Lie groups are rooted in this area.In greater detail, given a finite-dimensional vector space V of dimension n we can consider the symmetric algebraS(Sr(V)) of the polynomials of degree r over V, and the action on it of GL(V). It is actually more accurate to considerthe relative invariants of GL(V), or representations of SL(V), if we are going to speak of invariants: that is becausea scalar multiple of the identity will act on a tensor of rank r in S(V) through the r-th power 'weight' of the scalar.The point is then to define the subalgebra of invariants I(Sr(V)) for the action. We are, in classical language, lookingat invariants of n-ary r-ics, where n is the dimension of V. (This is not the same as finding invariants of GL(V) onS(V); this is an uninteresting problem as the only such invariants are constants.) The case that was most studied wasinvariants of binary forms where n = 2.Other work included that of Felix Klein in computing the invariant rings of finite group actions on C2 (the binarypolyhedral groups, classified by the ADE classification); these are the coordinate rings of du Val singularities.Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century,is once again at the forefront of mathematics.Kung & Rota (1984, p.27)

The work of David Hilbert, proving that I(V) was finitely presented in many cases, almost put an end to classicalinvariant theory for several decades, though the classical epoch in the subject continued to the final publications ofAlfred Young, more than 50 years later. Explicit calculations for particular purposes have been known in moderntimes (for example Shioda, with the binary octavics).

9.3 Hilbert’s theorems

Hilbert (1890) proved that if V is a finite-dimensional representation of the complex algebraic group G = SLn(C)then the ring of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used theReynolds operator ρ from R to RG with the properties

• ρ(1) = 1

• ρ(a + b) = ρ(a) + ρ(b)

• ρ(ab) = a ρ(b) whenever a is an invariant.

9.4. GEOMETRIC INVARIANT THEORY 23

Hilbert constructed the Reynolds operator explicitly using Cayley’s omega process Ω, though now it is more commonto construct ρ indirectly as follows: for compact groups G, the Reynolds operator is given by taking the average overG, and non-compact reductive groups can be reduced to the case of compact groups using Weyl’s unitarian trick.Given the Reynolds operator, Hilbert’s theorem is proved as follows. The ring R is a polynomial ring so is graded bydegrees, and the ideal I is defined to be the ideal generated by the homogeneous invariants of positive degrees. ByHilbert’s basis theorem the ideal I is finitely generated (as an ideal). Hence, I is finitely generated by finitely manyinvariants of G (because if we are given any – possibly infinite – subset S that generates a finitely generated ideal I,then I is already generated by some finite subset of S). Let i1,...,in be a finite set of invariants of G generating I (asan ideal). The key idea is to show that these generate the ring RG of invariants. Suppose that x is some homogeneousinvariant of degree d > 0. Then

x = a1i1 + ... + a i

for some aj in the ring R because x is in the ideal I. We can assume that aj is homogeneous of degree d − deg ij forevery j (otherwise, we replace aj by its homogeneous component of degree d − deg ij; if we do this for every j, theequation x = a1i1 + ... + ani will remain valid). Now, applying the Reynolds operator to x = a1i1 + ... + ani gives

x = ρ(a1)i1 + ... + ρ(an)in

We are now going to show that x lies in the R-algebra generated by i1,...,in.First, let us do this in the case when the elements ρ(ak) all have degree less than d. In this case, they are all in theR-algebra generated by i1,...,in (by our induction assumption). Therefore x is also in this R-algebra (since x = ρ(a1)i1+ ... + ρ(a )i ).In the general case, we cannot be sure that the elements ρ(ak) all have degree less than d. But we can replace eachρ(ak) by its homogeneous component of degree d − deg ij. As a result, these modified ρ(ak) are still G-invariants(because every homogeneous component of a G-invariant is a G-invariant) and have degree less than d (since deg ik> 0). The equation x = ρ(a1)i1 + ... + ρ(a )i still holds for our modified ρ(ak), so we can again conclude that x liesin the R-algebra generated by i1,...,in.Hence, by induction on the degree, all elements of RG are in the R-algebra generated by i1,...,in.

9.4 Geometric invariant theory

The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the constructionof a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtletheory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In aseparate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, hasbeen rehabilitated.One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing markedobjects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology,and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.

9.5 See also

• Gram’s theorem

• invariant theory of finite groups

• representation theory of finite groups

• Molien series

• invariant (mathematics)

24 CHAPTER 9. INVARIANT THEORY

9.6 References[1] Borel, Armand (2001). Essays in the History of Lie groups and algebraic groups. History of Mathematics, Vol. 21.

American mathematical society and London mathematical society. ISBN 978-0821802885.

• Dieudonné, Jean A.; Carrell, James B. (1970), “Invariant theory, old and new”, Advances in Mathematics 4:1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525 Reprinted as Dieudonné, Jean A.;Carrell, James B. (1971), “Invariant theory, old and new”, Advances in Mathematics (Boston, MA: AcademicPress) 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

• Dolgachev, Igor (2003), Lectures on invariant theory, London Mathematical Society Lecture Note Series 296,Cambridge University Press, doi:10.1017/CBO9780511615436, ISBN 978-0-521-52548-0, MR 2004511

• Grace, J. H.; Young, Alfred (1903), The algebra of invariants, Cambridge: Cambridge University Press

• Grosshans, Frank D. (1997), Algebraic homogeneous spaces and invariant theory, New York: Springer, ISBN3-540-63628-5

• Kung, Joseph P. S.; Rota, Gian-Carlo (1984), “The invariant theory of binary forms”, American MathematicalSociety. Bulletin. New Series 10 (1): 27–85, doi:10.1090/S0273-0979-1984-15188-7, ISSN 0002-9904, MR722856

• Hilbert, David (1890), “Ueber die Theorie der algebraischen Formen”, Mathematische Annalen 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831

• Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen 42 (3):313, doi:10.1007/BF01444162

• Neusel, Mara D.; and Smith, Larry (2002), Invariant Theory of Finite Groups, Providence, RI: AmericanMathematical Society, ISBN 0-8218-2916-5 A recent resource for learning about modular invariants of finitegroups.

• Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2 An undergraduate level introduction to the classical theory of invariants of binary forms, includingthe Omega process starting at page 87.

• Popov, V.L. (2001), “Invariants, theory of”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Springer, T. A. (1977), Invariant Theory, New York: Springer, ISBN 0-387-08242-5 An older but still usefulsurvey.

• Sturmfels, Bernd (1993), Algorithms in Invariant Theory, New York: Springer, ISBN 0-387-82445-6 A beau-tiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbnerbases.

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255

• Weyl, Hermann (1939b), “Invariants”, Duke Mathematical Journal 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030

9.7 External links• H. Kraft, C. Procesi, Classical Invariant Theory, a Primer

Chapter 10

The Classical Groups

In Weyl’s wonderful and terrible1 book The Classical Groups [W] one may discern two main themes: first, the studyof the polynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classicalgroup action; second, the isotypic decomposition of the full tensor algebra for such an action.1Most people who know the book feel the material in it is wonderful. Many also feel the presentation is terrible. (Theauthor is not among these latter.)Howe (1989, p.539)

In mathematics, The Classical Groups: Their Invariants and Representations is a book by Weyl (1939), whichdescribes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interestin invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.Weyl (1939b) gave an informal talk about the topic of his book.

10.1 Contents

Chapter I defines invariants and other basic ideas and describes the relation to Felix Klein's Erlanger program ingeometry.Chapter II describes the invariants of the special and general linear group of a vector space V on the polynomials overa sum of copies of V and its dual. It uses the Capelli identity to find an explicit set of generators for the invariants.Chapter III studies the group ring of a finite group and its decomposition into a sum of matrix algebras.Chapter IV discusses Schur–Weyl duality between representations of the symmetric and general linear groups.Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal andsymplectic groups, showing that the ring of invariants is generated by the obvious ones.Chapter VII describes the Weyl character formula for the characters of representations of the classical groups.Chapter VIII on invariant theory proves Hilbert’s theorem that invariants of the special linear group are finitely gen-erated.Chapter IX and X give some supplements to the previous chapters.

10.2 References

• Howe, Roger (1988), "The classical groups and invariants of binary forms”, in Wells, R. O. Jr., The mathe-matical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math. 48, Providence, R.I.:American Mathematical Society, pp. 133–166, ISBN 978-0-8218-1482-6, MR 974333

• Howe, Roger (1989), “Remarks on classical invariant theory.”, Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR

25

26 CHAPTER 10. THE CLASSICAL GROUPS

2001418, MR 0986027

• Jacobson, Nathan (1940), “Book Review: The Classical Groups”, Bulletin of the American Mathematical So-ciety 46 (7): 592–595, doi:10.1090/S0002-9904-1940-07236-2, ISSN 0002-9904, MR 1564136

• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255

• Weyl, Hermann (1939), “Invariants”, Duke Mathematical Journal 5: 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030

10.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 27

10.3 Text and image sources, contributors, and licenses

10.3.1 Text• Bracket algebra Source: https://en.wikipedia.org/wiki/Bracket_algebra?oldid=646373285 Contributors: Michael Hardy, TakuyaMurata,

Rjwilmsi, Michael Slone, SmackBot, Cronholm144, Geometry guy, Nilradical, Sun Creator, Yobot, AnomieBOT, Citation bot, Citationbot 1 and Deltahedron

• Bracket ring Source: https://en.wikipedia.org/wiki/Bracket_ring?oldid=646373328Contributors: TakuyaMurata, Andrewman327, R.e.b.,TexasAndroid, Headbomb, David Eppstein, AnomieBOT, AdventurousSquirrel and Deltahedron

• Canonizant Source: https://en.wikipedia.org/wiki/Canonizant?oldid=665953899Contributors: Michael Hardy, R.e.b., Trappist themonk,K9re11 and Anonymous: 1

• Capelli’s identity Source: https://en.wikipedia.org/wiki/Capelli’s_identity?oldid=644260317 Contributors: Michael Hardy, CharlesMatthews, Alan Liefting, Giftlite, Bender235, Woohookitty, Rjwilmsi, R.e.b., Headbomb, David Eppstein, JL-Bot, Mild Bill Hiccup,Addbot, Yobot, Citation bot, Citation bot 1, Darij, Alexander Chervov, RjwilmsiBot, John of Reading, K9re11 and Anonymous: 9

• Catalecticant Source: https://en.wikipedia.org/wiki/Catalecticant?oldid=492018302 Contributors: Michael Hardy, Rjwilmsi, R.e.b.,Headbomb and Citation bot

• Cayley’sΩprocess Source: https://en.wikipedia.org/wiki/Cayley’s_%CE%A9_process?oldid=599029089Contributors: Michael Hardy,Gene Ward Smith, Anthony Appleyard, Salix alba, R.e.b., Beetstra, Headbomb, Magioladitis, JackSchmidt, Citation bot 1, AlexanderChervov and Anonymous: 1

• Chevalley–Iwahori–Nagata theorem Source: https://en.wikipedia.org/wiki/Chevalley%E2%80%93Iwahori%E2%80%93Nagata_theorem?oldid=648032681 Contributors: TakuyaMurata, R.e.b., Wavelength, TexasAndroid, Headbomb, David Eppstein, Yobot, BattyBot, Doc-torKubla and K9re11

• Chevalley–Shephard–Todd theorem Source: https://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem?oldid=635233489 Contributors: Michael Hardy, Giftlite, Gro-Tsen, Rjwilmsi, R.e.b., Closedmouth, Headbomb, Magioladitis, Geometryguy, Arcfrk, Yobot, Citation bot, Citation bot 1, Brad7777, K9re11 and Anonymous: 4

• Invariant theory Source: https://en.wikipedia.org/wiki/Invariant_theory?oldid=661885946Contributors: Michael Hardy, CharlesMatthews,Michael Larsen, Rvollmert, Giftlite, Rgdboer, Ligulem, R.e.b., Ground Zero,Wavelength, Hillman, SmackBot, Polyade, Nbarth, Ligulem-bot, Headbomb, Sherbrooke, David Eppstein, Vegasprof, The enemies of god, Arcfrk, Alexbot, Addbot, Lightbot, PV=nRT, Javanbakht,Yobot, Citation bot, Omnipaedista, Dijkschneier, Darij, Jonesey95, TobeBot, Maxdlink, ClueBot NG, EdwardH, Mark L MacDonald,Stephan Alexander Spahn, Brirush, E E Ballew and Anonymous: 9

• The Classical Groups Source: https://en.wikipedia.org/wiki/The_Classical_Groups?oldid=665137034 Contributors: Michael Hardy,Tobias Bergemann, JIP, Rjwilmsi, R.e.b., Epbr123, Headbomb, David Eppstein, Trappist the monk and Anonymous: 2

10.3.2 Images• File:E-to-the-i-pi.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/35/E-to-the-i-pi.svg License: CC BY 2.5 Contribu-tors: ? Original artist: ?

• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Merge-arrow.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Merge-arrow.svg License: Public domain Con-tributors: ? Original artist: ?

• File:Mergefrom.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0f/Mergefrom.svg License: Public domain Contribu-tors: ? Original artist: ?

• File:Rubik’s_cube_v3.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b6/Rubik%27s_cube_v3.svg License: CC-BY-SA-3.0 Contributors: Image:Rubik’s cube v2.svg Original artist: User:Booyabazooka, User:Meph666 modified by User:Niabot

10.3.3 Content license• Creative Commons Attribution-Share Alike 3.0