Robinson–Schensted–Knuth Correspondence and Weak Polynomial Identities of M1,1(E)

2005 ◽  
Vol 12 (02) ◽  
pp. 333-349
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Roberto La Scala
Keyword(s):  
Hilbert Series ◽  
Quotient Algebra ◽  
Infinite Field ◽  
Polynomial Identities ◽  
The Ideal

In this paper, it is proved that the ideal Iw of the weak polynomial identities of the superalgebra M1,1(E) is generated by the proper polynomials [x1, x2, x3] and [x2, x1] [x3, x1] [x4, x1]. This is proved for any infinite field F of characteristic different from 2. Precisely, if B is the subalgebra of the proper polynomials of F<X>, we determine a basis and the dimension of any multihomogeneous component of the quotient algebra B / (B ∩ Iw). We also compute the Hilbert series of this algebra. One of the main tools of this paper is a variant we found of the Robinson–Schensted–Knuth correspondence defined for single semistandard tableaux of double shape.

GRADED IDENTITIES FOR THE ALGEBRA OF n×n UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD

2003 ◽  
Vol 13 (05) ◽  
pp. 517-526 ◽  
Author(s):  
PLAMEN KOSHLUKOV ◽  
ANGELA VALENTI
Keyword(s):  
Infinite Field ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

We consider the algebra Un(K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤn-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra.

On the factorability of the ideal of ⁎-graded polynomial identities of minimal varieties of PI ⁎-superalgebras

2022 ◽  
Vol 589 ◽  
pp. 273-286
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Viviane Ribeiro Tomaz da Silva ◽  
Ernesto Spinelli
Keyword(s):  
Polynomial Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

Cocharacters, Codimensions and Hilbert Series of the Polynomial Identities for 2 × 2 Matrices with Involution

1994 ◽  
Vol 46 (4) ◽  
pp. 718-733 ◽  
Author(s):  
Vesselin Drensky ◽  
Antonio Giambruno
Keyword(s):  
Hilbert Series ◽  
Polynomial Identities

AbstractLet M2(K, *) be the algebra of 2 × 2 matrices with involution over a field K of characteristic 0. We obtain the exact values of the cocharacters, codimensions and Hilbert series of the *-T-ideal of the polynomial identities for M2(K, *).

scholarly journals The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

2011 ◽  
Vol 141 (3) ◽  
pp. 609-629 ◽  
Author(s):  
Natalia Iyudu ◽  
Stanislav Shkarin
Keyword(s):  
Commutative Algebra ◽  
Hilbert Series ◽  
Infinite Field ◽  
Asymptotic Results ◽  
Quadratic Algebras ◽  
Princeton University ◽  
Number Of Generators ◽  
Finite Dimensional ◽  
Generic Algebra ◽  
Generic Algebras

We study the question of whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (non-commutative) algebra, is attained. This question was considered by Anick in his 1983 paper, ‘Generic algebras and CW-complexes’ (Princeton University Press), where he proved that the estimate is attained for the number of quadratic relations d ≤ ¼n2 and d ≥ ½n2, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to ½n(n – 1) was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional.We prove that over any infinite field, the Anick conjecture holds for d ≥ (n2 + n) and an arbitrary number of generators n, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

scholarly journals $q,t$-Catalan Numbers and Generators for the Radical Ideal defining the Diagonal Locus of $({\mathbb C}^2)^n$

10.37236/645 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li
Keyword(s):  
Vector Space ◽  
Hilbert Series ◽  
Upper Bounds ◽  
Catalan Numbers ◽  
Catalan Number ◽  
Minimal Set ◽  
Radical Ideal ◽  
The Ideal

Let $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the bi-graded vector space $M(=\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\text{dim }M_{d_1, d_2}$ in terms of number of partitions, and find all bi-degrees $(d_1,d_2)$ such that $\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$.

scholarly journals Graded polynomial identities for matrices with the transpose involution over an infinite field

2017 ◽  
Vol 46 (4) ◽  
pp. 1630-1640
Author(s):  
Luís Felipe Gonçalves Fonseca ◽  
Thiago Castilho de Mello
Keyword(s):  
Infinite Field ◽  
Polynomial Identities ◽  
Graded Polynomial Identities

scholarly journals The Hilbert series of rings of matrix concomitants

1988 ◽  
Vol 111 ◽  
pp. 143-156 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Characteristic Zero ◽  
Hilbert Series ◽  
Polynomial Identities

Throughout this paper, K will be a field of characteristic zero. Let K ‹x1,…, xm › be the K-algebra in m variables x1…, xm and Im, n the T-ideal consisting of all polynomial identities satisfied by m n by n matrices. The ring R(n, m) = K ‹x1,…, xm ›/Im, n is called the ring of m generic n by n matrices.

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