Invariant polynomial

In mathematics , an invariant polynomial is a polynomial on a vector space (see Symmetrical Algebra ), which is invariant under the action of a group on the vector space , i.e. ${\displaystyle P}$ ${\displaystyle G}$ ${\displaystyle V}$

${\displaystyle P(gx)=P(x)}$

fulfilled for all . ${\displaystyle g\in G,x\in V}$

Invariant Polynomials in Linear Algebra

Let be a field and the vector space of all matrices over . The general linear group acts through conjugation: ${\displaystyle \mathbb {K} }$ ${\displaystyle V=Mat(n,\mathbb {K} )}$ ${\displaystyle n\times n}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle GL(n,\mathbb {K} )}$ ${\displaystyle V}$

${\displaystyle gx:=gxg^{-1}}$for . ${\displaystyle g\in GL(n,\mathbb {K} ),x\in Mat(n,\mathbb {K} )}$

In this case, invariant polynomials are functions with for everyone . ${\displaystyle P:Mat(n,\mathbb {K} )\rightarrow \mathbb {K} }$ ${\displaystyle P(gxg^{-1})=P(x)}$ ${\displaystyle g\in GL(n,\mathbb {K} ),x\in Mat(n,\mathbb {K} )}$

Examples are the trace and the determinant of matrices. More generally one can (with a formal variable ) the development ${\displaystyle t}$

${\displaystyle det(tA+I)=\sum _{k=0}^{n}c_{k}(A)t^{k}}$

consider and get invariant polynomials . ( is the trace and the determinant. If is algebraically closed , then in general is the kth elementary symmetric polynomial in the eigenvalues of .) ${\displaystyle c_{0},\ldots ,c_{n}}$ ${\displaystyle c_{1}}$ ${\displaystyle c_{n}}$ ${\displaystyle \mathbb {K} }$ ${\displaystyle c_{k}}$ ${\displaystyle A}$

Invariant polynomials in the theory of Lie groups

Let be a Lie group and its Lie algebra . A polynomial on is a polynomial (with real coefficients) in the basis vectors of , see Symmetric Algebra . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$

The group acts on itself through conjugation: for everyone . The differential of is a linear map ${\displaystyle G}$ ${\displaystyle c_{g}(h):=ghg^{-1}}$ ${\displaystyle h\in G}$ ${\displaystyle c_{g}}$

${\displaystyle Ad(g):=D(c_{g})_{e}:{\mathfrak {g}}\rightarrow {\mathfrak {g}}}$,

this defines the so-called adjoint representation of the group on the vector space . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$

An invariant polynomial is a polynomial which is invariant under the adjoint effect, that is ${\displaystyle {\mathfrak {g}}}$

${\displaystyle P(Ad(g)X_{1},\ldots ,Ad(g)X_{k})=P(X_{1},\ldots ,X_{k})}$ for all ${\displaystyle g\in G,X_{1},\ldots ,X_{k}\in {\mathfrak {g}}}$

Fulfills. The algebra of the invariant polynomials is denoted by. ${\displaystyle I^{*}({\mathfrak {g}})}$

Example GL ( n , ℝ)

In this case is and for . For is the homogeneous polynomial of degree , whose value is given as coefficient of degree in the polynomial ${\displaystyle {\mathfrak {g}}=Mat(n,\mathbb {R} )}$ ${\displaystyle Ad(g)(A)=gAg^{-1}}$ ${\displaystyle g\in GL(n,\mathbb {R} ),A\in Mat(n,\mathbb {R} )}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle P_{\frac {k}{2}}}$ ${\displaystyle k}$ ${\displaystyle (A,\ldots ,A)}$ ${\displaystyle n-k}$

${\displaystyle det(\lambda \mathbb {I} -{\frac {1}{2\pi }}A)=\sum _{k}P_{\frac {k}{2}}(A,\ldots ,A)\lambda ^{n-k}}$

receives, for everyone . (The values ​​for these clearly define a polynomial.) The polynomial is called the -th Pontryagin polynomial . ${\displaystyle A\in Mat(n,\mathbb {R} )}$ ${\displaystyle (A,\ldots ,A)}$ ${\displaystyle P_{\frac {k}{2}}}$ ${\displaystyle {\frac {k}{2}}}$

The algebra of the invariant polynomials is generated by the . ${\displaystyle P_{\frac {k}{2}}\in I^{k}({\mathfrak {g}}l(n,\mathbb {R} ))}$

Example G = O ( n )

For holds , from which it follows first and then for all odd ones. ${\displaystyle A\in {\mathfrak {o}}(n)}$ ${\displaystyle A=-A^{T}}$ ${\displaystyle det(\lambda \mathbb {I} -{\frac {1}{2\pi }}A)=det(\lambda \mathbb {I} +{\frac {1}{2\pi }}A)}$ ${\displaystyle P_{\frac {k}{2}}=0}$ ${\displaystyle k}$

The algebra of the invariant polynomials is generated by the . ${\displaystyle P_{k}\in I^{2k}({\mathfrak {o}}(n))}$

Example G = SO ( n )

If is even, one also has Pfaff's determinant , which is defined for with by ${\displaystyle n=2m}$ ${\displaystyle A=(a_{ij})}$ ${\displaystyle a_{ij}=-a_{ji}}$

${\displaystyle Pf(A,\ldots ,A)={\frac {1}{2^{2m}\pi ^{m}m!}}\sum _{\sigma \in S_{2m}}sign(\sigma )a_{\sigma (1)\sigma (2)}\ldots a_{\sigma (2m-1)\sigma (2m)}}$.

The algebra of the invariant polynomials is generated by the Pontryagin polynomials and - if is even - the Pfaff determinant (also known as the Euler polynomial ) . ${\displaystyle P_{k}\in I^{2k}({\mathfrak {s}}o(n))}$ ${\displaystyle n}$ ${\displaystyle Pf\in I^{\frac {n}{2}}({\mathfrak {s}}o(n))}$

Example G = GL ( n , ℂ)

For let the complex-valued homogeneous polynomial be of degree , whose value is given as coefficient of degree in the polynomial ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle C_{\frac {k}{2}}}$ ${\displaystyle k}$ ${\displaystyle (A,\ldots ,A)}$ ${\displaystyle n-k}$

${\displaystyle det(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A)=\sum _{k}C_{k}(A,\ldots ,A)\lambda ^{n-k}}$

receives, for everyone . The polynomial is called the -th Chern polynomial . The Chern and Pontryagin polynomials are related by the equation . ${\displaystyle A\in Mat(n,\mathbb {C} )}$ ${\displaystyle C_{k}}$ ${\displaystyle k}$ ${\displaystyle i^{k}C_{k}(A,\ldots ,A)=P_{\frac {k}{2}}(A,\ldots ,A)}$

The algebra of the complex-valued invariant polynomials is generated by the . ${\displaystyle C_{k}\in I^{k}({\mathfrak {g}}l(n,\mathbb {C} ))}$

Example G = U ( n )

For is and therefore therefore the Chern polynomials are real-valued. ${\displaystyle A\in {\mathfrak {u}}(n)}$ ${\displaystyle A=-A^{H}}$ ${\displaystyle det\left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)={\overline {det\left(\lambda \mathbb {I} -{\frac {1}{2\pi i}}A\right)}},}$ ${\displaystyle {\mathfrak {u}}(n)}$

The algebra of the invariant polynomials is generated by the . ${\displaystyle C_{k}\in I^{k}({\mathfrak {u}}(n))}$

literature

• Shoshichi Kobayashi , Katsumi Nomizu : Foundations of differential geometry. Vol. I, II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969.
• Johan L. Dupont : Curvature and characteristic classes. Lecture Notes in Mathematics, Vol. 640. Springer-Verlag, Berlin-New York, 1978. ISBN 3-540-08663-3