I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., …

I am looking for references (both of the readable and of the historical kind!) for the following result (which I formulate in one of its least general forms, so as not to complicat …

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+ …

We know that an important variant of the Schur Polynomial is the shifted Schur Polynomials that was developed by Okounkov & Olshanski. The question here is: is there any other …

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$. General question. Does there exist a simple …

Background: For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric funct …

I'm looking for a reference for the following fact. Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic …

Some will recognize this as similar to a question I asked before, but I want to ask it without the trigonometry. Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $ …

The question is in the title, however; there is a Hopf algebra of quasisymmetric functions which has the Hopf algebra of symmetric functions as a sub - Hopf algebra. The quasisymme …

$\newcommand{\lm}{\lambda}$ $\newcommand{\bR}{\mathbb{R}}$ Let $m$ be an integer $>1$. Define $$ I_m:\bR^m\to \bR,\;\; I_m(\lm_1,\dotsc, \lm_m)=\int_{S^{m-1}}\exp\Bigl(-\sum_{j= …

For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$ one has Jacobian, expressed by the $(n \times n)$-determinants: $$ J(f_1,\dots,f_n) …

What is known about S_n group invariants in a free associative (noncommutative) algebra k < x_1, ...x_n >) ? (S_n natural acts by permutations on generators). What is Poincare …

I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov ch …

Hello All, I warn in advance my math skills are quite limited so speaking to me like I am an idiot would be appreciated. I have a polynomial : $y = a\cdot e^{\frac{-2(x - x_0)^2 …

I'm wondering if there is a characterization for whether $$p_\lambda(x_1, \dotsc, x_r) \geq p_\mu (x_1, \dotsc, x_r) \text{ for every $r \in \mathbb{N}$ and $x_1, \dotsc, x_r \in \ …