standard monomials
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10.37236/9874 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Chanchal Kumar ◽  
Gargi Lather ◽  
Sonica

 Let $G$ be a graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro were the first to consider a monomial ideal $\mathcal{M}_G$, called the $G$-parking function ideal, in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively to $G$-parking functions. Dochtermann introduced and studied skeleton ideals of the graph $G$, which are subideals of the $G$-parking function ideal with an additional parameter $k ~(0\le k \le n-1)$. A $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of the graph $G$ is generated by monomials corresponding to non-empty subsets of the set of non-root vertices $[n]$ of size at most $k+1$. Dochtermann obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the $k$-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.



2020 ◽  
Vol 343 (4) ◽  
pp. 111785
Author(s):  
Tamás Mészáros


2019 ◽  
Vol 75 (3) ◽  
pp. 584-592 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Dong-il Lee

For the group algebra of the finite non-crystallographic Coxeter group of type H 4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.



2019 ◽  
Vol 19 (01) ◽  
pp. 2050002
Author(s):  
Sung Soon Kim ◽  
Dong-il Lee
Keyword(s):  

For Temperley–Lieb algebras of types [Formula: see text] and [Formula: see text], we construct their Gröbner-Shirshov bases and the corresponding standard monomials



Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 438
Author(s):  
Jeong-Yup Lee ◽  
Dong-il Lee ◽  
SungSoon Kim

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra T ( d , n ) of the complex reflection group G ( d , 1 , n ) , inducing the standard monomials expressed by the generators { E i } of T ( d , n ) . This result generalizes the one for the Coxeter group of type B n in the paper by Kim and Lee We also give a combinatorial interpretation of the standard monomials of T ( d , n ) , relating to the fully commutative elements of the complex reflection group G ( d , 1 , n ) . More generally, the Temperley-Lieb algebra T ( d , r , n ) of the complex reflection group G ( d , r , n ) is defined and its dimension is computed.



2018 ◽  
Vol 17 (02) ◽  
pp. 1850037
Author(s):  
Ajay Kumar ◽  
Chanchal Kumar

For an (oriented) graph [Formula: see text] on the vertex set [Formula: see text] (rooted at [Formula: see text]), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal [Formula: see text] in the polynomial ring [Formula: see text] over a field [Formula: see text] such that the number of standard monomials of [Formula: see text] equals the number of (oriented) spanning trees of [Formula: see text] and hence, [Formula: see text], where [Formula: see text] is the truncated Laplace matrix of [Formula: see text]. The standard monomials of [Formula: see text] correspond bijectively to the [Formula: see text]-parking functions. In this paper, we study a monomial ideal [Formula: see text] in [Formula: see text] having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal [Formula: see text] is the cellular resolution supported on a subcomplex of the first barycentric subdivision [Formula: see text] of an [Formula: see text] simplex [Formula: see text]. The integer sequence [Formula: see text] has many interesting properties. In particular, we obtain a formula, [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], similar to [Formula: see text].



2017 ◽  
Vol 61 ◽  
pp. 855-861
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai


2017 ◽  
Vol 50 (4) ◽  
pp. 179-181
Author(s):  
SungSoon Kim ◽  
Dong-il Lee
Keyword(s):  


2016 ◽  
Vol 15 (08) ◽  
pp. 1650146 ◽  
Author(s):  
Dong-il Lee

For the exceptional Weyl group of type [Formula: see text], its Gröbner–Shirshov basis and the corresponding standard monomials are constructed.



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