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.

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.

ALGEBRAS SATISFYING THE POLYNOMIAL IDENTITY [x1,x2][x3,x4,x5]=0

2004 ◽  
Vol 03 (02) ◽  
pp. 121-142 ◽  
Author(s):  
ONOFRIO M. DI VINCENZO ◽  
VESSELIN DRENSKY ◽  
VINCENZO NARDOZZA
Keyword(s):  
Polynomial Identity ◽  
Characteristic Zero ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Proper Subvariety

Let [Formula: see text] be a field of characteristic zero, and [Formula: see text] the variety of associative unitary algebras defined by the polynomial identity [x1,x2][x3,x4,x5]=0. This variety is one of the several minimal varieties of exponent 3 (and all proper subvarieties are of exponents 1 and 2). We describe asymptotically its proper subvarieties. More precisely, we define certain algebras ℛ2k for any k∈ℕ and show that if [Formula: see text] is a proper subvariety of [Formula: see text], then the T-ideal of its polynomial identities is asymptotically equivalent to the T-ideal of the identities of one of the algebras [Formula: see text], E, ℛ2k or ℛ2k⊕E, for a suitable k∈ℕ. We give also another description relating the T-ideals of the proper subvarieties of [Formula: see text] with the polynomial identities of upper triangular matrices of a suitable size.

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, *).

GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS

2007 ◽  
Vol 06 (03) ◽  
pp. 385-401 ◽  
Author(s):  
ONOFRIO M. DI VINCENZO ◽  
VINCENZO NARDOZZA
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Characteristic Zero ◽  
Grassmann Algebra ◽  
Polynomial Identities ◽  
Graded Algebras ◽  
Infinite Dimensional ◽  
Graded Polynomial Identities ◽  
New Type

Let F be a field and let E be the Grassmann algebra of an infinite dimensional F-vector space. For any p,q ∈ ℕ, the algebra Mp,q(E) can be turned into a ℤp+q × ℤ2-algebra by combining an elementary ℤp+q-grading with the natural ℤ2-grading on E. The tensor product Mp,q(E) ⊗ Mr,s(E) can be turned into a ℤ(p+q)(r+s) × ℤ2-algebra in a similar way. In this paper, we assume that F has characteristic zero and describe a system of generators for the graded polynomial identities of the algebras Mp,q(E) and Mp,q(E) ⊗ Mr,s(E) with respect to these new gradings. We show that this tensor product is graded PI-equivalent to Mpr+qs,ps+qr(E). This provides a new proof of the well known Kemer's PI-equivalence between these algebras. Then we classify all the graded algebras Mp,q(E) having no non-trivial monomial identities, and finally calculate how many non-isomorphic gradings of this new type are available for Mp,q(E).

Codimension growth of central polynomials of Lie algebras

2020 ◽  
Vol 32 (1) ◽  
pp. 201-206
Author(s):  
Antonio Giambruno ◽  
Mikhail Zaicev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Adjoint Representation ◽  
Closed Field ◽  
Polynomial Identities ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Linearly Independent

AbstractLet L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero and let I be the T-ideal of polynomial identities of the adjoint representation of L. We prove that the number of multilinear central polynomials in n variables, linearly independent modulo I, grows exponentially like {(\dim L)^{n}}.

scholarly journals COCHARACTERS OF POLYNOMIAL IDENTITIES OF UPPER TRIANGULAR MATRICES

2012 ◽  
Vol 11 (01) ◽  
pp. 1250018 ◽  
Author(s):  
SILVIA BOUMOVA ◽  
VESSELIN DRENSKY
Keyword(s):  
Explicit Form ◽  
Generating Function ◽  
Characteristic Zero ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices

Let T(Uk) be the T-ideal of the polynomial identities of the algebra of k × k upper triangular matrices over a field of characteristic zero. We give an easy algorithm which calculates the generating function of the cocharacter sequence χn(Uk) = Σλ⊢n mλ(Uk)χλ of the T-ideal T(Uk). Applying this algorithm we have found the explicit form of the multiplicities mλ(Uk) in two cases: (i) for the "largest" partitions λ = (λ1,…,λn) which satisfy λk+1 +⋯+ λn = k - 1; (ii) for the first several k and any λ.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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