All Questions
Tagged with ra.rings-and-algebras co.combinatorics
99 questions
9
votes
3
answers
1k
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Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
6
votes
1
answer
199
views
Combinatorial type construction of the free operad
$\DeclareMathOperator\RT{RT}$I am reading the book "Algebraic operads" by J. L. Loday and B. Vallete. The authors have given a combinatorial construction of the free operad over an $\mathbb{...
- 229
3
votes
2
answers
468
views
How fast does the number of "fixed" points grow compared to the size of the ball in the following group?
I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...
- 823
1
vote
0
answers
83
views
Non-vanishing of product of zero divisors in quotients modulo $n$
This might be of practical importance and even partial answer will help.
Let $n$ be odd squarefree integer with known factorization $n=\prod p_i$
with $N$ prime factors.
Later we are not asking about ...
- 25.4k
2
votes
1
answer
124
views
Proof of dynamic programming calculation of Levenshtein distance
Let s1 and s2 are 2 arbitrary strings with lengths l1 and ...
- 203
1
vote
0
answers
189
views
The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 173
6
votes
1
answer
231
views
Does Manin's construction of non-commutative endomorphism algebra $\mathrm{End}(A)$ produce Koszul algebra, if $A$ is Koszul?
$\newcommand{\dual}{\mathrm{dual}}\DeclareMathOperator\End{End}\DeclareMathOperator\Fun{Fun}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\GL{GL}$Around 1986–7 Yu.Manin proposed natural and ...
- 24.8k
2
votes
2
answers
77
views
Reference request for a subfamily of regular graphs
[Repost of same question math stack exchange which got no answers]
I'm looking for literature on the following family of graphs:
Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
- 21
4
votes
1
answer
211
views
Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
- 10.6k
2
votes
1
answer
298
views
Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
- 557
19
votes
2
answers
851
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The discriminant of the Okada algebra
The Okada algebra $\mathfrak{O}_n$ over a field $K$ has generators
$E_1,\dots,E_{n-1}$ and relations $E_i^2=x_iE_i$,
$E_{i+1}E_iE_{i+1}=y_i E_{i+1}$, and $E_iE_j=E_jE_i$ for $|i-j|\geq
2$, where $x_i,...
- 50.8k
0
votes
0
answers
116
views
Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 10.5k
1
vote
0
answers
118
views
An algebra with two multiplications, based on series-parallel diagrams?
Here is a commutative, unital, associative algebra $\mathcal{F}$ with two ways to multiply. The multiplications come from a construction with Boolean operations and series-parallel diagrams. I want ...
- 1,277
2
votes
0
answers
114
views
How many minimal relations are needed to obtain a Frobenius algebra?
Let $A_n:=K \langle x_1,x_2,...,x_n \rangle$ be the non-commutative polynomial ring in $n$-variables over the field $K$ and let $J=\langle x_1,...,x_n \rangle$ be the ideal spanned by the $x_i$.
An ...
- 26.5k
4
votes
1
answer
370
views
Determining when quotient of a polynomial ring is a Gorenstein ring
I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
- 214
3
votes
0
answers
116
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A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)
Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
- 10.5k
11
votes
9
answers
1k
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What are examples of problems we know how to solve for primes (or prime powers), but not for composites?
I am interested in seeing examples of research problems which fall into one of the two following categories:
A problem which is solved in the case of primes (or prime powers), but which remains open ...
4
votes
0
answers
259
views
Road map for learning cluster algebras
I'm a PhD student and I would like learn about cluster algebras. I'm wondering what is a good reference (i.e., has detailed explanations, examples, etc) to learn from the basic and what are some of ...
- 839
2
votes
0
answers
102
views
When do two path algebras share an underlying graph?
Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction.
Since ...
- 433
7
votes
1
answer
193
views
Free median algebras and maximal linked systems
$\DeclareMathOperator\MLS{MLS}$Recall that the median operation, on the power set $2^Y$ of subsets of a set $Y$, is the ternary law $m(A,B,C)$ mapping a triple of subsets to the set of elements ...
- 63.9k
1
vote
0
answers
329
views
Outlier absences of monomials in a group of inversion partition polynomials
Revamped and updated on Sep 12, 2022:
Given the complex coefficients $a_n$ of some generic formal power, Taylor, Laurent or other series, say the ordinary generating functions (o.g.f.) $f(z) = z +a_1 ...
- 10.5k
6
votes
1
answer
135
views
Automorphisms of special egg-box diagrams
By a egg-box diagram I will simply mean a (possibly infinite) rectangular array of holes, with some of the holes containing an egg (denoted by a filled-in circle) and the rest of the holes are empty (...
- 18.7k
4
votes
0
answers
143
views
Non-Desarguesian finite projective planes with ≤3 (non-collinear) chosen points, and coordinatisation
It is well-known that an arbitrary projective plane can have very different symmetry group to a field plane. In particular, the symmetries are not transitive on the set of fundamental quadrangles. ...
- 35.4k
9
votes
2
answers
789
views
Algebraic power series of finite order
Apologies if the question is too elementary/something well-known.
I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under ...
- 24.2k
10
votes
2
answers
882
views
The maximal subset of a finite field where the sum of any subset is non-zero
Given a finite field $\mathbb{F}_q$ with $q=p^m$ where $p$ is the characteristic.
For any subset $S=\{a_1,\dots,a_n\}$ of $\mathbb{F}_q$, if any partial sum (i.e. the sum of elements in a non-empty ...
- 441
0
votes
1
answer
97
views
On polynomials associated to integers power sums [closed]
For $0\leq k\leq n$ integers let $P_k(n):= n^k,\ S_k(n):= P_k(1)+\ldots P_k(n)= 1^k+\ldots n^k$.
Then $P_k(0)=0$, $S_0(n)=n$.
For calculate $S_1(n)$ i consider:
$$P_2(n)-P_2(n-1)=2n+1$$
then
$\begin{...
- 4,165
10
votes
1
answer
272
views
Plane partitions as irreducible representations
The irreducible representations of the symmetric group algebras $A_n=KS_n$ over a the complex numbers (or any field of characteristic 0) $K$ satisfy the following properties:
The irreducible ...
- 26.5k
8
votes
1
answer
238
views
Functions over monoids which factor in two different ways
This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
- 32.1k
4
votes
1
answer
451
views
Non-associative commutative "group"
When dealing with some hash functions that I was trying to speed up, I toyed with a binary operation with the goal to "approximate" the addition on $\{0,1\}^*$ when seen as binary ...
- 48.8k
11
votes
1
answer
329
views
Are there three non-commutative polynomials in three variables with finite dimensional quotient?
$\newcommand\la{\langle}\newcommand\ra{\rangle}$Let $K$ be a field and $K\la x,y,z\ra$ the non-commutative polynomial ring in 3 variables.
Question 1: Are there three (fewer is probably not possible?!...
- 26.5k
2
votes
0
answers
63
views
Finding non-commutative finite-dimensional "hypersurface" algebras
Fix a field $K$.
Call a non-commutative polynomial $f(x_i)$ whose monomial terms are all of degree at least 2 in the variables $x_i$ magic if the finite dimensional $K$-algebra $A_{f,n}:=K<x_i>/(...
- 26.5k
7
votes
2
answers
589
views
On a matrix problem in the field $\mathbb F_2$
Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...
- 13.9k
1
vote
0
answers
142
views
How can I build free unital magmas?
N. Bourbaki formally defines the free magma $M(X)$ over a set $X$. However, it does not define the free unital magma over $X$, which I am denoting by $M^{\ast}(X)$ (maybe you know some more common ...
- 353
3
votes
0
answers
107
views
Do Frobenius algebras have a lattice basis and what lattices do appear?
Let $K$ be for simplicity be the field with two or three elements (or alternatively we could restrict to ideals containing only the field elements $-1$ or $1$ as coefficients).
A (commutative) ...
- 26.5k
26
votes
2
answers
2k
views
Function of $(x_1,x_2,x_3,x_4)$ that factors in two ways as $\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Suppose we have a function $f(x_1 ,x_2 ,x_3 ,x_4).$ We know that we can factor it in two ways as $f(x_1 ,x_2 ,x_3 ,x_4)=\phi_1 (x_1 ,x_2 )\phi_2(x_3 ,x_4 )=\psi_1 (x_1,x_3)\psi_2(x_2,x_4)$
Show that ...
- 519
5
votes
0
answers
139
views
How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
- 765
10
votes
1
answer
807
views
How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
12
votes
1
answer
584
views
Unit group of octonions over finite fields
One can define the algebra $A(K)$ of octonions over an arbitrary field $K$, see for example the command OctaveAlgebra in GAP: https://www.gap-system.org/Manuals/doc/ref/chap62.html .
When $K$ is a ...
- 26.5k
32
votes
1
answer
2k
views
Is this formal noncommutative power series identity known?
I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...
- 114k
4
votes
0
answers
153
views
The Jacobson radical as a bimodule
Let $A$ be a finite dimensional algebra with Jacobson radical $J$.
Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten ...
- 26.5k
5
votes
0
answers
345
views
A fusion ring identity
Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
- 309
2
votes
0
answers
72
views
injective map between tensor products of two irreducible modules of simple Lie algebra sl_{n+1}
Let $1 \leq i_1 < i_2 < i_3 \leq n$. I know that there is an injective map from $V(\omega_{i_1}+\omega_{{i_2} -1})\otimes V(\omega_{{i_3}+1})$ to $V(\omega_{i_1}+\omega_{i_2})\otimes V(\omega_{...
- 137
7
votes
1
answer
566
views
Is there an integral fusion ring which is not of Frobenius type?
Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...
4
votes
1
answer
236
views
Solving equations in the Brauer algebra
(First asked in MSE)
The Brauer algebra $B_n(x)$ is an algebra of matchings whose product is described here.
Given $A$ and $B$ two elements of $B_n(x)$, and given an integer $m$, there are in ...
- 1,549
3
votes
1
answer
158
views
Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]
I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...
- 1,677
4
votes
0
answers
70
views
Number of ideals in an algebra
Let $R_{n,m}^q$ be the finite dimensional algebra $K\langle x_1,...,x_n\rangle/J^m$, where the field $K$ has $q$ elements and $K\langle x_1,...,x_n\rangle$ is the non-commutative polynomial ring with ...
- 26.5k
2
votes
1
answer
229
views
Has the "semidirect monoid of a semiring" been considered anywhere?
Given a semiring $S$, we get a monoid $M(S)$ as follows:
The underlying set of $S$ is $S^2$
The identity element is $(0,1)$
The law of composition is given by $$(a,A)(b,B) = (Ba+b,AB),$$ where $a,A,b$...
- 3,793
1
vote
1
answer
139
views
Is the Cartan permanent odd for finite global dimension?
Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix.
Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd ...
- 26.5k
11
votes
0
answers
450
views
A congruence involving roots of unity
Let $f(x) \in \mathbb{Z}[x]$ and suppose $f(\omega^j) \in \mathbb{Z}$ for all $j= 1, \dots, n$ where $\omega = e^{2 \pi i/n}$ is a primitive $n^{\text{th}}$ root of unity.
Computational evidence ...
- 305
6
votes
2
answers
309
views
Permanent of Nakayama algebras
See https://en.wikipedia.org/wiki/Nakayama_algebra for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix.
(all algebras are ...
- 26.5k