Search results for "Triangular matrix"
showing 10 items of 28 documents
POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH
2001
Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.
Gradings on the algebra of upper triangular matrices and their graded identities
2004
Abstract Let K be an infinite field and let UT n ( K ) denote the algebra of n × n upper triangular matrices over K . We describe all elementary gradings on this algebra. Further we describe the generators of the ideals of graded polynomial identities of UT n ( K ) and we produce linear bases of the corresponding relatively free graded algebras. We prove that one can distinguish the elementary gradings by their graded identities. We describe bases of the graded polynomial identities in several “typical” cases. Although in these cases we consider elementary gradings by cyclic groups, the same methods serve for elementary gradings by any finite group.
Two Reflected Gray Code-Based Orders on Some Restricted Growth Sequences
2014
We consider two order relations: that induced by the m-ary reflected Gray code and a suffix partitioned variation of it. We show that both of them when applied to some sets of restricted growth sequences still yield Gray codes. These sets of sequences are: subexcedant and ascent sequences, restricted growth functions and staircase words. In particular, we give the first suffix partitioned Gray codes for restricted growth f unctions and ascent sequences; these latter sequences code various combinatorial classes as interval orders, upper triangular matrices without zero rows and zero columns whose non-negative integer entries sum up to n, and certain pattern-avoiding permutations. For each Gr…
Minimal star-varieties of polynomial growth and bounded colength
2018
Abstract Let V be a variety of associative algebras with involution ⁎ over a field F of characteristic zero. Giambruno and Mishchenko proved in [6] that the ⁎-codimension sequence of V is polynomially bounded if and only if V does not contain the commutative algebra D = F ⊕ F , endowed with the exchange involution, and M , a suitable 4-dimensional subalgebra of the algebra of 4 × 4 upper triangular matrices , endowed with the reflection involution. As a consequence the algebras D and M generate the only varieties of almost polynomial growth. In [20] the authors completely classify all subvarieties and all minimal subvarieties of the varieties var ⁎ ( D ) and var ⁎ ( M ) . In this paper we e…
Polynomial growth and star-varieties
2016
Abstract Let V be a variety of associative algebras with involution over a field F of characteristic zero and let c n ⁎ ( V ) , n = 1 , 2 , … , be its ⁎-codimension sequence. Such a sequence is polynomially bounded if and only if V does not contain the commutative algebra F ⊕ F , endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4 × 4 upper triangular matrices. Such algebras generate the only varieties of ⁎-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the ⁎-varieties of almost polynomial growth by gi…
Gradings on the algebra of upper triangular matrices of size three
2013
Abstract Let UT 3 ( F ) be the algebra of 3 × 3 upper triangular matrices over a field F . On UT 3 ( F ) , up to isomorphism, there are at most five non-trivial elementary gradings and we study the graded polynomial identities for such gradings. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . We finally compute the multiplicities in the graded cocharacter sequence for every elementary G -grading on UT 3 ( F ) .
The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
2018
We compute systematically for the planar double box Feynman integral relevant to top pair production with a closed top loop the Laurent expansion in the dimensional regularisation parameter $\varepsilon$. This is done by transforming the system of differential equations for this integral and all its sub-topologies to a form linear in $\varepsilon$, where the $\varepsilon^0$-part is strictly lower triangular. This system is easily solved order by order in the dimensional regularisation parameter $\varepsilon$. This is an example of an elliptic multi-scale integral involving several elliptic sub-topologies. Our methods are applicable to similar problems.
GRADED IDENTITIES FOR THE ALGEBRA OF n×n UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD
2003
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.
Superinvolutions on upper-triangular matrix algebras
2018
Let UTn(F) be the algebra of n×n upper-triangular matrices over an algebraically closed field F of characteristic zero. In [18], the authors described all abelian G-gradings on UTn(F) by showing that any G-grading on this algebra is an elementary grading. In this paper, we shall consider the algebra UTn(F) endowed with an elementary Z2-grading. In this way, it has a structure of superalgebra and our goal is to completely describe the superinvolutions which can be defined on it. To this end, we shall prove that the superinvolutions and the graded involutions (i.e., involutions preserving the grading) on UTn(F) are strictly related through the so-called superautomorphisms of this algebra. We …
Varieties of special Jordan algebras of almost polynomial growth
2019
Abstract Let J be a special Jordan algebra and let c n ( J ) be its corresponding codimension sequence. The aim of this paper is to prove that in case J is finite dimensional, such a sequence is polynomially bounded if and only if the variety generated by J does not contain U J 2 , the special Jordan algebra of 2 × 2 upper triangular matrices. As an immediate consequence, we prove that U J 2 is the only finite dimensional special Jordan algebra that generates a variety of almost polynomial growth.