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.

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.

scholarly journals Special Cases of the Parking Functions Conjecture and Upper-Triangular Matrices

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Paul Levande
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Parking Functions ◽  
Combinatorial Interpretation ◽  
Forthcoming Article ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Diagonal Harmonics ◽  
Special Cases ◽  
Bivariate Generating Function

International audience We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article. On examine les cas spéciaux $q=1$ et $t=0$ de la conjecture des fonctions de stationnement. Cette conjecture déclare que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à la fonction génératrice bivariée (paramètres $area$ et $dinv$) sur l'ensemble des fonctions de stationnement. Haglund a prouvé récemment que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à une fonction génératrice bivariée sur l'ensemble des matrices de Tesler triangulaires supérieures dont la somme de chaque équerre vaut un. On donne une interprétation combinatoire de la fonction génératrice de Haglund pour $q=1$ et on prouve le cas correspondant de la conjecture dans le cas des fonctions de stationnement (prouvé d'abord par Garsia et Haiman). On discute aussi d'une preuve possible du cas $t=0$, cohérente avec cette interprétation combinatoire. On conclut en discutant brièvement les raffinements possibles de la conjecture des fonctions de stationnement de ce point de vue. $\textbf{Note ajoutée sur épreuve}$: j'ai trouvé depuis cet article une preuve du cas $t=0$ et conjecturé des raffinements possibles. Ces résultats seront probablement présentés dans un article ultérieur.

Graded Involutions on Upper-triangular Matrix Algebras

2009 ◽  
Vol 16 (01) ◽  
pp. 103-108 ◽  
Author(s):  
A. Valenti ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Upper Triangular Matrices ◽  
Upper Triangular Matrix

Let UTn be the algebra of n × n upper-triangular matrices over an algebraically closed field of characteristic zero. We describe all G-gradings on UTn by a finite abelian group G commuting with an involution (involution gradings).

scholarly journals Counting self-dual interval orders

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vít Jelínek
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Interval Orders ◽  
Maximal Elements ◽  
Triangular Matrices ◽  
Bijective Correspondence ◽  
Upper Triangular Matrices ◽  
Positive Entry ◽  
International Audience ◽  
Natural Statistics

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

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.

scholarly journals Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

2021 ◽  
Author(s):  
Onofrio M. Di Vincenzo ◽  
Vincenzo Nardozza
Keyword(s):  
Lie Algebra ◽  
Differential Polynomial ◽  
Two Dimensional ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Generating Set ◽  
Upper Triangular Matrices

AbstractWe study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

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