**40**

votes

**1**answer

3k views

### Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...

**0**

votes

**0**answers

70 views

### Action of the (special) orthogonal group on differential forms

I was told that the following facts are true. I am looking for a reference to them.
1) The action of $O(n,\mathbb{C})$ on $\wedge^l\mathbb{C}^n$ is irreducible for any $l$.
2) The action of ...

**7**

votes

**1**answer

294 views

### Quotients by the additive group $\mathbb G_a$

Geometric invariant theory doesn't work so well for non-reductive groups, since invariant rings are not generally finitely generated. However, in many cases the action of a non-reductive group has a ...

**5**

votes

**2**answers

202 views

### Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function
$P_i = px^i + qy^i + rz^i$
where $x,y,z$ are coordinates. I have a few related ...

**2**

votes

**4**answers

385 views

### Equivalent binary forms

Two binary forms $f, g \in k[x, y]$ are equivalent when there exists an $M \in GL_2 (k)$ such that $f^M = g$. For simplicity we take $k$ such that $char (k) =0$ and $k=\bar k$.
The equivalence ...

**4**

votes

**1**answer

213 views

### Calculation with weights of $E_6$

Question: Consider the complex simple Lie group $E_6$. Let $\lambda_1$ and $\lambda_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^*$, resp. Consider the complex ...

**3**

votes

**0**answers

121 views

### Invariants of the symmetric group

Let $V_\lambda$ be an irreducible representation of the symmetric group $S_n$ as usual labeled by parition $\lambda$ of $n.$
Question.
Is there any general information about the algebra of ...

**5**

votes

**0**answers

110 views

### Generators for invariant tensors of lie algebras

EDITED FOR (hopeful) CLARITY:
For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...

**5**

votes

**2**answers

188 views

### Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...

**8**

votes

**2**answers

500 views

### Representations of $SL(2)$ in characteristic 2

In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of SL(2)-modules. In characteristic $p$ things are more complicated.
I am interested in the special case ...

**3**

votes

**4**answers

339 views

### Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...

**6**

votes

**3**answers

388 views

### $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows
$$
(A,B) \cdot (M_1, \ldots, M_m) ...

**15**

votes

**2**answers

999 views

### Invariants for the exceptional complex simple Lie algebra $F_4$

This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.
Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...

**5**

votes

**1**answer

112 views

### Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?

In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...

**0**

votes

**0**answers

18 views

### function invariants under integral transforms

Some integral transforms have various invariants (Fourier transform, etc.), while others don't, such as Hilbert transform. I am wondering if there is any general method to find the invariant functions ...

**5**

votes

**0**answers

65 views

### Invariant theory of $SL_2$ over a field of positive characteristic

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...

**1**

vote

**1**answer

184 views

### Schur-Weyl duality for arbitrary tensor products of simple finite-dimensional $GL_n$-modules

Let $M$ and $N$ be two simple finite dimensional $GL_n$ modules. Is there a way of expressing the heighest weight vectors of the simple submodules of $M\otimes N$ in terms of the heighest weight ...

**8**

votes

**0**answers

159 views

### Earliest use of the term “linearly reductive”?

Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...

**3**

votes

**0**answers

84 views

### First Fundamental Theorem for Alternating Group

I know it fails but is there an answer?
More precisely, let $V$ be the standard complex $n$-dimensional representation of the alternating group $A_n$, $kV$ the direct sum of its $k$ copies, $S(kV)$ ...

**0**

votes

**0**answers

120 views

### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...

**2**

votes

**0**answers

90 views

### The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request

All cohomology and homology will be $Z/2$ coefficient. The restriction map
$H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of
the Dickson invariant $Z/2[w_2,w_3]$ into the ...

**1**

vote

**0**answers

75 views

### Galois correspondence for action of general linear group on purely transcendental extension

For a fixed positive integer $n$, the group $G=GL_n(\mathbb{C})$ acts on the field $K=\mathbb{C}(t_1,\ldots,t_n)$ by linear change of variables. I would like to know if there is something like a ...

**4**

votes

**1**answer

217 views

### The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...

**4**

votes

**1**answer

221 views

### Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...

**1**

vote

**1**answer

112 views

### Specht polynomials for dihedral groups

The representation theory of the symmetric group is best understood via the Specht polynomials. In wonder how this works for other finite reflection groups, such as dihedral groups. Are the similarly ...

**2**

votes

**0**answers

72 views

### Weyl group invariants of the representation ring of a split torus

Let $G$ be a semisimple split algebraic group, $T$ its split maximal torus and $W$ corresponding Weyl group. Let $T^*$ denote the character lattice of $T$ and $\Lambda$ denote the weight lattice, so ...

**0**

votes

**2**answers

140 views

### Decomposition of $G$-harmonic polynomials

Let $G$ be a finite group and let $H_G$ denote the $G$-harmonic polynomials. What is the structure of $H_G$ as a $G$-module? Is it isomorphic to the regular representation?

**7**

votes

**1**answer

169 views

### Chevalley restriction theorem for exterior algebras

Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem ...

**7**

votes

**2**answers

371 views

### Equivariant normalization?

Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...

**4**

votes

**2**answers

289 views

### Quotient of affine space by cyclic permutation

The quotient of the affine space $\mathbb{A}^n$ by the symmetric group $Sym_n$ is again an affine space of the same dimension, and invariants are given by elementary symmetric polynomials.
What ...

**2**

votes

**0**answers

54 views

### different and discriminant for finite invariants

Let $k$ be an algebraically closed field.
Let $B$ a $k$-algebra of finite type, normal and Cohen-macaulay. Let $G$ a finite group acting on $B$. We assume that the order of $G$ is prime to the ...

**3**

votes

**1**answer

264 views

### quotient by finite group actions that are smooth

Let $X$ an affine normal scheme of finite type over a field $k$ of characteristic zero.
Let $G$ a finite group acting on $X$ and $Y=X/G=Spec(K[X]^{G})$.
We assume that ...

**3**

votes

**0**answers

135 views

### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...

**4**

votes

**1**answer

353 views

### Where does the name “Reynolds operator” come from?

I always found it strange that, in the context of invariant and representation theory,
averaging over the group is called the "Reynolds operator". As far as I know the work of Reynolds was in fluid ...

**0**

votes

**0**answers

143 views

### Multiplicative groups of fields

Let $G$ be a finite group, and let be a field on which $G$ acts
faithfully. Let $C$, be a $\mathbb{Z}G$-lattice, and let $\alpha$
be a one-cocycle from $G$ to $Hom(C,L^{*})$. Let $L[C]$ the group
...

**9**

votes

**2**answers

301 views

### Extension of the Weyl dimension formula

Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...

**1**

vote

**1**answer

68 views

### Jacobian of primary invariants

Let $G$ be a finite group and consider $R=K[V]^G$ be the invariant ring of the group G over the field K of char 0. Let $f_1,\ldots f_n$ be a set of primary invariants. Is there a nice geometric ...

**2**

votes

**0**answers

196 views

### How to determine there exists a unique invariant subspace for a set of matrices

Hi everyone,
Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...

**1**

vote

**1**answer

543 views

### What is the “fundamental theorem of invariant theory” ?

The basic question I guess can be formulated as - given two integers $N_f$ and $N_c$ what are the ways in which the fundamental and the anti-fundamental representations of $U(N_f)$ be combined to get ...

**4**

votes

**3**answers

178 views

### Generate a higher degree symmetric polynomial from an existing one

Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as
$q(x_1, x_2, \cdots, x_{n+1}) = ...

**4**

votes

**2**answers

244 views

### Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?
E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...

**0**

votes

**1**answer

113 views

### Ideal membership (concerning polynomial invariants of orthogonal groups)

Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials
$$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$
I'm ...

**1**

vote

**1**answer

104 views

### Can one pick generators for the ring of invariants of binary n-ic forms which have rational coefficients?

The problem of determining a set of generators of the ring of invariants of the group $\textrm{SL}_2$ acting on the complex $n+1$-dimensional vector space of binary $n$-ic forms is known to be very ...

**2**

votes

**1**answer

207 views

### Invariants of a set of real unit vectors in 3d space, under SO(3)

I have a set of $n$ real unit vectors, in 3-dimensional space.
(It is a follow up of Sets of vectors related by a rotation.)
Is there a construction providing a complete set of independent*) ...

**1**

vote

**0**answers

60 views

### An almost permutation G-lattice

I've been trying to determine the rationality of certain fields of invariants coming from G-lattices. More precisely, letting $G$ be a finite group, $L=\mathbb{Z}^n$ a free abelian group with a $G$ ...

**1**

vote

**0**answers

74 views

### A slightly odd (integral of Whittaker functions / sum of characters of $GL_n(\mathbb C)$ / sum of Schur functions)

Let $W$ be the normalized spherical Whittaker function attached to a spherical representation $\pi$ on $GL_n(k)$, where $k$ is a $p$-adic field and $n\ge 3$.
I'm faced with the slightly-odd integral
...

**5**

votes

**0**answers

143 views

### algebra of endomorphisms over the diagonal invariants

Let $k$ be a field of characteristic 0 (say $\mathbb{C}$).
Consider the ring of polynomials $R = k[X_1,...,X_n]$ and its subring of invariant polynomials $S = R^{S_n}$.
It is known that the ...

**2**

votes

**1**answer

158 views

### Modules of invariants?

Let $G \subset SL(V, \mathbb{C})$ be a finite group and $R=(\operatorname{Sym}\[V\])^G$ is the ring of polynomial invariants, $W$ some irreducible complex representation of $G$. I want to know is ...

**1**

vote

**1**answer

227 views

### What does this particular geometric quotient locally look like?

Let $k$ be a field and consider the algebraic group $GL_n$ over $Spec(k)$. It has as a closed (but not normal) algebraic subgroup the group $M$ of monomial matrices, i.e. matrices having exactly one ...

**2**

votes

**1**answer

134 views

### Invariance group of Morse charts

Suppose I have a smooth function $\varphi$ that vanishes at $p$ and has a positive definite Hessian at that point (suppose that we are on a smooth manifold of dimension $M$). Then the Morse lemma ...