Tight monomials for type B3

The global crystal basis or canonical basis plays an important role in the theory of the quantized enveloping algebras and their representations. The tight monomials are the simplest elements in the canonical basis. We discuss the tight monomials in quantized enveloping algebra of type B3.     

A functional equation having monomials as solutions

For each n=1,2,3, we obtain the general solution and the stability of the functional equation

f(2x+y)+f(2x-y)=2^n-2[f(x+y)+f(x-y)+6f(x)].     

Tightness of certain monomials

For a quantized enveloping algebra of finite type, one can associate a natural monomial to a dominant weight. We show that these monomials for types A5 and D4 are semitight(i.e., a Z-linear combination of elements in the canonical basis) by a direct calculation.     

On the real rank of monomials

In this paper we study the real rank of monomials and we give an upper bound for it. We show that the real and the complex ranks of a monomial coincide if and only if the least exponent is equal to one.

     

Hyperplane arrangement cohomology and monomials in the exterior algebra

We show that if is the complement of a complex hyperplane arrangement, then the homology of has linear free resolution as a module over the exterior algebra on the first cohomology of . We study invariants of that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.

     

Primitive rings with involution and pivotal monomials

With the aid of ultrafilters an example is constructed of a primitive ring with involution and zero socle in which the symmetric elementsS satisfys ² c=s,c depending on and commuting withs (thusS satisfies the pivotal monomialx). However, by suitably restricting the definition of generalized pivotal monomial (essentially by making the variables linear) it is shown that a primitive ring with involution has nonzero socle if and only if the symmetric elements satisfy a restricted GPM.     

Moment functions and exponential monomials on commutative hypergroups

Aequationes mathematicae - The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its...     

Inequalities for linear combinations of monomials in p-Newton sequences

The partial order on monomials that corresponds to domination when evaluated at positive Newton sequences is fully understood. Here we take up the corresponding partial order on linear combinations of monomials. In part using analysis based upon the cone structure of the exponents in p-Newton sequences, an array of conditions is given for this new partial order. It appears that a characterization in general will be difficult. Within the case in which all coefficients are 1, the situation in which, for general sequence length, there are two monomials, each of length two and nonnegative integer exponents, the partial order is fully characterized. The characterization is combinatorial, in terms of indices in the monomials, and, already here there is much more than term-wise domination.     

Desingularization of Quiver Grassmannians via Nakajima Categories

In this paper, we show that generalized Nakajima Categories provide a framework to construct a desingularization of quiver Grassmannians for self-injective algebras of finite representation type. Furthermore, we show that all standard Frobenius models of orbit categories of the bounded derived category considered in Keller, Documenta Math. 10: 551–581, 2005 are equivalent to proj ??, the finitely generated projective modules of the regular Nakajima category ??.     

Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofx ^k by polynomials of degreen (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann ² and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p _k,n , which can be identified with a certain probability. The numberp _k,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.     

Algebraic gamma monomials and double coverings of cyclotomic fields

We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex , to compute , where is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension , we define a double covering of to be an extension of degree , such that is Galois. We demonstrate that each class gives rise to a double covering of , by . When lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of is abelian and hence . The may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

     

Toric rings generated by special stable sets of monomials

In this paper we consider some subalgebras of the d-th Veronese subring of a polynomial ring, generated by stable subsets of monomials. We prove that these algebras are Koszul, showing that the presentation ideals have Gröbner bases of quadrics with respect to suitable term orders. Since the initial monomials of the elements of these Gröbner bases are square- free, it follows by a result of STURMFELS [S, 13.15], that the algebras under consideration are normal, and thus Cohen-Macaulay.     

Complexity of certain systems of monomials in calculation by composition circuits

The circuit complexity of monomial set computation is studied in the paper. In the model considered here, the complexity means the minimal number of composition operations sufficient for calculating the system from its variables. It is established that the considered complexity measure can be much less than known complexity measures corresponding to models admitting, for example, either multiplication operations only, or multiplication and division operations, or multiplication operations with the ability to use inverse variables. However, this feature of significant “computation strength” is not universal, which is confirmed by an appropriate example. Furthermore, for a system containing two monomials of two variables we obtained an exact complexity value. We have also established that duality reasons do not work (or work poorly) in calculations using composition operation.     

共形几何代数中的分阶幂零单项式的几何解释

本文进一步发展共形几何代数,从代数和几何两个角度对共形几何代数中由幂零向量生成的分阶单项式的代数性质和几何解释进行深入探讨,包括各种维数的球面和平面定向的代数刻画、长括号的三角学、角度和方向的幂零单项式分阶表示、幂零单项式各个分阶的几何意义等.这些结果有助于理解高层次符号代数运算背后的几何意义,以及共形几何代数简化符号几何计算的内在机制.