5
votes
0answers
120 views

Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients. $$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$ Monomials $x^k$ are mapped to $n ...
4
votes
2answers
311 views

A question about pairs of lines in 3D projective space

Consider a 3-dimensional projective space $X$. Let $m$ be the smallest number so that there are $m$ pairs of lines $ \ell_1,\ell'_1$, $ \ell_2,\ell_2'$, ... , $\ell_m, \ell'_m$ in $X$: a) For ...
1
vote
0answers
44 views

On generating Euler Square of index q, q-1 (where q is any prime power)

Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...
11
votes
1answer
507 views

Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...
2
votes
1answer
111 views

A non-orthomodular orthocomplemented lattice identity?

Assume you have an orthocomplemented (but possibly not orthomodular) lattice $L$. For $q,r\in L$ say "$q$ and $r$ are in position $p'$" to mean that $q\wedge r^\perp=0$ and $r\wedge q^\perp=0$. Is ...
12
votes
0answers
512 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in Fran├žois Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
9
votes
4answers
532 views

zeros of a homogeneous polynomial

Hi All, Let $F$ be a finite field, $\lambda\in F$, and $$p_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ...
10
votes
2answers
284 views

Temperley-Lieb algebras for other Weyl groups?

The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
3
votes
0answers
247 views

det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n

Context: some probably know that there are Capelli identities which state det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th century, but also ...
12
votes
4answers
769 views

Showing that a family of polynomials has positive and real roots.

Hi everybody, for my research I am dealing with the following function: $$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in ...
10
votes
5answers
580 views

Is complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?

Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$. $$ h_a=\text{ sum of all monomials of degree ...
7
votes
1answer
320 views

Sequences without long arithmetic progressions

First, a bit of notation. If we have an arithmetic progression $a, a+k, a+2k, \ldots, a+(n-1)k$ we will call $k$ the distance, and $n$ the length. While trying to find an example for a paper I'm ...
3
votes
3answers
237 views

measuring n 2-planes in R^{2n}.

Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ ...
9
votes
1answer
206 views

Linearization of a fact about functions between finite sets

Suppose $X$, $Y$, and $Z$ are finite sets. If we have a function $$f : X \longrightarrow Y$$ and another $$g : Y \longrightarrow Z$$ then the composite function $g \circ f$ has the property that $$ ...
4
votes
2answers
294 views

monomials in the universal enveloping of a Lie algebra in terms of the symmetric basis

Let $L$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero and $e_1,\ldots, e_n$ some basis of $L$. The formula $[e_i,e_j] = \sum_k C_{ij}^k e_k$ determines the structure ...
6
votes
3answers
755 views

Which graphs are zero-divisor graphs for some ring?

Given a (non commutative) ring $R$, we construct a (directed) graph $G_0(R)$ with vertex set $Z(R)\backslash \{0\}$, the zero divisors of $R$ except for $0$. And an edge from $x$ to $y$ whenever ...
7
votes
1answer
365 views

The ring generated by all functions from a set to itself

Let $S$ be a finite set. Now $\mathop{End}(S)$ is a monoid, and we may build a ring $R$ by allowing formal sums of functions. Preliminary questions, since $R$ is surely well-known: What is it ...
1
vote
0answers
120 views

Bases of Ideals With no Monomials

Let $K$ be an algebraically closed field and $K[\underline{x}]$ its ring of polynomials in $n$ variables $x_1,\cdots, x_n$. Let $J\leq K[\underline{x}]$ be an ideal such that there are no monomials in ...
1
vote
0answers
315 views

The sum of a nilpotent left ideal and a nil left ideal

In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal. I am having ...
2
votes
0answers
122 views

Formal solutions of semiring equations

I am looking for a general theorem which would tell me when a formal series solution exists for an equation over a semiring. One may assume that the semiring is equipped with a (formal) derivative. ...
4
votes
0answers
347 views

Monomial-type ideals in polynomial rings

Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
14
votes
3answers
2k views

Proving that a poset is a lattice

I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably ...
2
votes
1answer
500 views

Smith Normal Form and lower triangular Toeplitz Matrices

I am working on a undergrad research project with some other guys. Now the conjecture (unrelated to this question) we are trying to prove boils down to a final subproblem: Let $A (n \times n)$ be a ...
3
votes
0answers
240 views

Algebraic Kneser conjecture?

Recall that Kneser conjecture (now Lovasz theorem) claims that if the family of $k$-subsets (subsets of cardinality $k$) of given $(2k+d)$-set $M$, $d\geq 1$ are colored into $d+1$ colors, then there ...