Number Of Positive Roots Research Articles

Root cycles in Coxeter groups

Abstract For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

Descartes’ rule of signs, Rolle’s theorem and sequences of compatible pairs

AbstractConsider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r+,r−), where r+ is the number of positive roots and r− the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r+,r−) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for , where (resp.) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.

Representation growth of compact linear groups

We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(O). This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is r/κ, where r is the rank and κ the number of positive roots. We then show that the twist zeta function of GLn(O) exists and has the same abscissa of convergence as the zeta function of SLn(O), provided n does not divide char O. We compute the twist zeta function of GL2(O) when the residue characteristic p of O is odd and approximate the zeta function when p = 2 to deduce that the abscissa is 1. Finally, we construct a large part of the representations of SL2(Fq[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2].

Quality of positive root bounds

In this paper, we study the quality of positive root bounds. A positive root bound of a polynomial is an upper bound on the largest positive root. Higher quality means that the relative over-estimation (the ratio of the bound and the largest positive root) is smaller. We report three findings.(1)Most known positive root bounds can be arbitrarily bad; that is, the relative over-estimation can approach infinity, even when the degree and the coefficient size are fixed.(2)When the number of sign variations is the same as the number of positive roots, the relative over-estimation of a positive root bound due to Hong (BH) is at most linear in the degree, no matter what the coefficient size is.(3)When the number of sign variations is one, the relative over-estimation of BH is at most constant, in particular 4, no matter what the degree and the coefficient size are.

SL2-factorizations of Chevalley groups

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bezout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

We construct a linear representation of the CGW algebra of type Dn. This representation has degree n2 - n, the number of positive roots of a root system of type Dn. We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type Dn, this representation is equivalent to the faithful linear representation of Cohen–Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type Dn. Our proof is computer-assisted using Mathematica.

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.

More on the Lost Cousin of the Fundamental Theorem of Algebra

g(x) = a0x0 + a1x1 + · · · + an xn , where α0 < α1 < · · · < αn are arbitrary real numbers and an = 0, has no more than n roots. Proof. We proceed by induction on n, noting that for n = 1 the statement is obvious. Assume that for some n the claim is true, but for n + 1, it is not. Hence, for some real numbers α0 < α1 < · · · < αn < αn+1 and an+1 = 0, there is a function g(x) = a0x0 + a1x1 + · · · + an xn + an+1xn+1 , whose number of positive roots is larger than n + 1. These roots are identical with the roots of the new function

A Note on Exponents vs Root Heights for Complex Simple Lie Algebras

We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra $\mathfrak{ g}$, the partition formed by the exponents of $\mathfrak{ g}$ is dual to that formed by the numbers of positive roots at each height.

New Designs for the Descartes Rule of Signs

Descartes's rule of signs says that the number of positive roots of p(x) is equal to the number of sign changes in the sequence ao, al, ... , an, or is less than this number by a positive even integer. Investigating which of the possible numbers of roots permitted by Descartes actually occur, Anderson, Jackson, and Sitharam [1] show that the rule cannot be improved. For any sign sequence not containing zero, they construct polynomials with this sign sequence in the coefficients and any prescribed number of positive roots that is in accord with Descartes's rule. Analogously, since the negative roots are the roots of p(-x), all allowable numbers of negative roots are realized. Grabiner [2] establishes further examples. In particular, he provides polynomials for sign sequences that may contain zero and shows how to achieve certain numbers of positive and negative roots simultaneously.

Representations of the nonstandard (twisted) deformation U′ q (so n ) for q a root of unity

Irreducible representations of the algebrasU′q(son) forq a root of unityqp=1 are given. The main class of these representations act onpN-dimensional linear space (whereN is a number of positive roots of the Lie algebra son) and are given byr = dim son complex parameters. Some classes of degenerate irreducible representations are also described.

Descartes' Rule of Signs: Another Construction

(1999). Descartes' Rule of Signs: Another Construction. The American Mathematical Monthly: Vol. 106, No. 9, pp. 854-856.

A Conjecture of Erdos

The modern formulation of Descartes' Rule of Signs provides a list of possible numbers of both positive and negative roots for a given sign sequence. We have shown (assuming none of the signs are zero) that polynomials exist with any of the possible numbers of positive roots. By replacing x by -x, we can provide a polynomial with the given sign sequence that contains any given number of negative roots allowable by Descartes. We have not addressed trying to accommodate both the positive and negative roots simultaneously. The following question is unanswered here and seems to be open: Given a sign sequence (which may include some zeros), do there exist polynomials containing positive and negative roots numbering each of the possible combinations allowed by Descartes' Rule of Signs?

The Kostant cone and the combinatorial structure of the root lattice

Let R be an irreducible root system with base B, weight lattice ,4, fundamental dominant weights D with respect to B, dominant weights /i + with respect to D, root lattice A,, and A,? the nonnegative integral sums of members of B. Then /i,’ C$ /i +, but A, c A. Let Z, be the nonnegative integers. Let x E AT. Then x has coordinates in Z; as a root sum with respect to B, and coordinates in Z” as a weight with respect to D. The difference of these coordinates y, in Z”, plays a key role in the study of Kostant’s partition function, the deeper combinatorial study, now required in various areas of mathematics and physics, of the representation theory of complex semisimple Lie algebras, and the modular version of this representation theory. If YEZ”,, i.e., with all nonnegative entries, then x is a sum of positive nonsimple roots, a member of the Kostant cone of A,?, so called because all the calculations of Kostant’s partition function, P, in A,? can be done in the Kostant cone, by means of an easy operator, s, on A,?, defined in terms of y. If y E z; , but with all strictly positive entries, then x is in the interior of the Kostant cone; otherwise it is on the boundary. For xE,4,f, let E+(x) be a realization of the partition function, i.e., the set of all expressions of x as a sum of positive roots. The coordinates of any such expression would be in Zy , where m is the number of positive roots of R. This E+(x) is rich in combinatorial structure, most notably i-blocks and zero blocks. The jth i-block of E+(x) is the set of expressions in E+(x) whose coefficient of the simple root ai is exactly j. The relations between the i-blocks inside of E+(x), using y, correspond to the relations between certain weight spaces in irreducible modules. Also, y provides the entering wedge into the crucial lowest i-blocks. The zero block of E+(x) is the intersection, over i, of all the zeroth i-blocks of E+(x), i.e., the expressions in 81 0021-8693/89 $3.00

Progress in Radio-Echo Sounding Theory

AbstractThe inverse problem of radio-echo sounding consists in the reconstruction of subglacial relief from the known radio-echo profile, and the path, and speed of the aircraft. The present work shows that in the geometrical optical approach the solution of the inverse problem for a homogeneous, two-dimensional object (valley glacier) exists, and there is a unique solution. An algorithm for interpretation of the experimental data is suggested. It may be considered as the generalization of Harrison’s transformations for any surfaces. paths, and speed.The direct problem of radio-echo sounding consists in the reconstruction of radio-echo profile from the known surface and subsurface relief, path, and aircraft speed. The analysis of traces in the standard radio-echo sounding mode of operation reveals the possibility of introducing a three-index trace classification {K, sign S’(o), K+} where K is the number of real roots characteristic of the equations, K+ the number of positive roots, and S’(o) the position derivative, the argument being equal to zero. The form {o, — o} is optimal for the precision of the calculation of the reflected surface coordinates, as well as for the simplicity of the interpreted picture. By a special choice of the altitude of flight, any form of any surface can be brought to {o, —o}. The decrease in beam width is equivalent to the diminution of roots of the characteristic equations. For a pencil beam the trace degenerates into a point.The attenuation of the reflected signal depends on the glacier geometry, the dielectric parameters of the medium, the altitude and the course of the aircraft, as well as statistical characteristics of the mutual orientation of the interfaces and aerials. For the description of the energetics of radio-echo sounding, equivalent reflecting surfaces are suggested. These surfaces correspond to the Harrison’s equivalent reflecting surface. Exact formulae for the power of coherent and incoherent components of the reflected signal are obtained. Components of the full attenuation such as absorption, depolarization, and spherical divergence, are investigated with respect to refraction and focusing effects.

Progress in Radio-Echo Sounding Theory

Abstract The inverse problem of radio-echo sounding consists in the reconstruction of subglacial relief from the known radio-echo profile, and the path, and speed of the aircraft. The present work shows that in the geometrical optical approach the solution of the inverse problem for a homogeneous, two-dimensional object (valley glacier) exists, and there is a unique solution. An algorithm for interpretation of the experimental data is suggested. It may be considered as the generalization of Harrison’s transformations for any surfaces. paths, and speed. The direct problem of radio-echo sounding consists in the reconstruction of radio-echo profile from the known surface and subsurface relief, path, and aircraft speed. The analysis of traces in the standard radio-echo sounding mode of operation reveals the possibility of introducing a three-index trace classification {K, sign S’(o), K+} where K is the number of real roots characteristic of the equations, K+ the number of positive roots, and S’(o) the position derivative, the argument being equal to zero. The form {o, — o} is optimal for the precision of the calculation of the reflected surface coordinates, as well as for the simplicity of the interpreted picture. By a special choice of the altitude of flight, any form of any surface can be brought to {o, —o}. The decrease in beam width is equivalent to the diminution of roots of the characteristic equations. For a pencil beam the trace degenerates into a point. The attenuation of the reflected signal depends on the glacier geometry, the dielectric parameters of the medium, the altitude and the course of the aircraft, as well as statistical characteristics of the mutual orientation of the interfaces and aerials. For the description of the energetics of radio-echo sounding, equivalent reflecting surfaces are suggested. These surfaces correspond to the Harrison’s equivalent reflecting surface. Exact formulae for the power of coherent and incoherent components of the reflected signal are obtained. Components of the full attenuation such as absorption, depolarization, and spherical divergence, are investigated with respect to refraction and focusing effects.

Weyl Groups, Deformations of Linkage Classes, and Character Degrees For Chevalley Groups

With a Weyl group W and a positive integer p are associated p-linkage classes of weights [4,13]. Small deformations of such classes by elements of W are introduced here. These lead in turn to certain polynomials in p with highest term pm, m = number of positive roots (one polynomial for each conjugacy class of W), which are written down explicitly for types A1, A2, B2. These polynomials give (for each prime p) the degrees of the various large series of irreducible characters of the corresponding Chevalley group over the field of p elements. Indeed, the formal behavior of weights appears to reflect the actual behavior of the characters under reduction modulo p.

Weyl ǵroups and finite Chevalley ǵroups

In his fundamental paper (1) Chevalley showed how to associate with each complex simple Lie algebra L and each field K a group G = L(K) which is (in all but four exceptional cases) simple. If K is a finite field GF(q), G is a finite group of orderwhere l is the rank of L, m is the number of positive roots of L and d is a certain integer determined by L and K. The integers m1, m2,…,m1 are determined by L only and satisfy the condition