**8**

votes

**0**answers

151 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 ...

**8**

votes

**0**answers

175 views

### Characteristic classes from moduli of alternating forms

Suppose, just for example, that you have a smooth manifold $M$ of dimension greater than $8$,and a cohomology class $q$ in $H^3(M,{\Bbb R})$. Suppose further that one can represent $q$ with a ...

**6**

votes

**0**answers

454 views

### Can one give a “nice” expression for this determinant?

I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague.
...

**5**

votes

**0**answers

58 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 ...

**5**

votes

**0**answers

141 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 ...

**5**

votes

**0**answers

253 views

### FFT like theorems for tensor product

The fundamental theorem of symmetric functions states that $\mathbb C[V]^{S_n}$ is generated by the elementary symmetric polynomials, where $V$ is the natural representation of $S_n$. Then a theorem ...

**5**

votes

**0**answers

192 views

### Modular reduction of exceptional complex reflection groups

I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that ...

**4**

votes

**0**answers

80 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) ...

**4**

votes

**0**answers

222 views

### Group action on Grassmannian: Intersection of two special invariant rings

Let $K$ be a field with characteristic $0$.
Let $G:=G(d,nd)$ the Grassmannian of all $d-$dimensional subspaces of $K^{nd}$ and let $H:=O_d(K)^n$ the n-fold direct product of the orthogonal group. $H$ ...

**4**

votes

**0**answers

309 views

### A relative Noether number for invariants

EDIT: Wrong definition of $\beta\left(G,H\right)$ fixed. One of the results is open (i. e., I cannot prove it).
In "Finite Groups and invariant theory" (a paper in Malliavin's LNM #1478 which can ...

**4**

votes

**0**answers

209 views

### When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?

Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements ...

**3**

votes

**0**answers

78 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, $S(kV)$ ...

**3**

votes

**0**answers

126 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 ...

**3**

votes

**0**answers

261 views

### Ring of invariants of finite subgroup of $GL_2(\mathbb{C})$

In the paper 'FINITE LINEAR GROUPS WHOSE RING OF INVARIANTS IS A COMPLETE INTERSECTION' by VICTOR KAC AND KEI-ICHI WATANABE published in BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY ...

**3**

votes

**0**answers

165 views

### Invariant Subvarieties of Variety of Quiver Representations

I'd like to understand a special case of the following rather general algebraic geometry question:
Given an algebraic group $G$ acting on a variety $V$, can we describe the $G$-invariant subvarieties ...

**2**

votes

**0**answers

77 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 ...

**2**

votes

**0**answers

66 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 ...

**2**

votes

**0**answers

48 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 ...

**2**

votes

**0**answers

186 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

**0**answers

70 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 ...

**1**

vote

**0**answers

58 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

72 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
...

**1**

vote

**0**answers

406 views

### Equivalence classes induced on binary strings by set of permutations

Let $\mathbb{F}_2^{n}$ be the set of binary strings of length $n$ and let $f: \mathbb{F}_2^{n} \rightarrow \mathbb{R}$ be a function from the set of binary strings of length $n$ to the reals.
Let's ...

**0**

votes

**0**answers

15 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 ...

**0**

votes

**0**answers

108 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 ...

**0**

votes

**0**answers

129 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
...

**0**

votes

**0**answers

27 views

### The invariant field for the dual of the augmentation lattice

Let $G$ be some finite group and $J_G$ be $ZG$ modulo the ideal $\{n\sum_{g} g | n\in\mathbb N\}$, or equivalently the dual of the augmentation ideal which is the kernel of $ZG\to Z$ sending $g$ to ...

**0**

votes

**0**answers

163 views

### Invariant Polynomes under group action - given the invariants looking for the group. algorithmic solution?

I have given a finite set $S$ of polynomes in the ring $R = C[x_1,\dots,x_n]$. I need to know the minimal group $G$ wich acts on $R$ such that $C[S]$ is the ring of invariants of $R$ under the action ...