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A result of Popov and Vinberg gives a positive case. For a simply-connected semi-simple algebraic group $G$ in characteristic zero. they consider the orbits of the highest weight vectors under in irr …

If we are given one particular 'factorization': $F_4 = F_3F_1-F_2^2$, we can proceed like this:- Express $F_2$ as product of irreducible polynomials, say, $F_2 = P_1^{m_1}P_2^{m_2}\cdots P_k^{m_k}$. …

For linear algebraic groups $G$, the picard group is the group of characters on $G$ which is finitely generated (and will be trivial for semi-simple groups). So one does not talk of moduli space of d …

A result of H.W.Lenstra (Inventiones Math., 1974): (a clearly written paper) For a prime number $p$ let $K=Q(x_1,x_2,\ldots, x_p)$, be a pure transcendental extension over the rational numbers. Let …

Orthogonal group consists of fixed-points of an involutary automorphism in $\operatorname{GL}(n)$. There is a general theory of involutions and symmetric varieties. For an involution $\sigma$ of a sem …

I have listened to lectures that mention Jonquières automorphisms for affine spaces by name. They don't seem to be found in textbooks on algebraic geometry. I would like to know the exact reference pr …

$O(n)$, by definition preserves the bilinear forms. Consider a subspace $W_r$ of dimension $r$ where the form is zero for any pair of vectors. Put it as the $r$th term of a complete flag $F$. Now con …

References for this purpose are: A series of papers initiated by C S Seshadri, Lakshmibai, Musili develops "Standard Monomial Theory" to deal with this. It gives equations for Schubert varietes, desc …

Don't know if there is a name. Possibly this is known to Newton; the inductive proof of Newton's theorem on elementary symmetric polynomials goes along similar lines. When we start with some polynom …

An excellent non-algebraic meaning (using analysis) of normality is found in Kollar's article in the Bulletin of AMS (1987). Restrict to irreducible varieties $X$ so we can talk of function fields. A …

First about $S_n$: when it acts by permutating the variable there is a nice description of invariants, as the transpositions will be represented by reflections. Chevalley's theorem states the invarian …

Finding sufficient conditions is wildly open territory; one can easily list conditions that are far far away from necessary. For example conditions that 0 is a root, or 1 is a root, or $-1$ is a ro …