We give the first analysis of a systematic scan version of the Metropolis algorithm. Our examples include generating random elements of a Coxeter group with probability determined by the length function. The analysis is based on interpreting Metropolis walks in terms of the multiplication in the Iwahori-Hecke algebra.persi diaconis and arun ram is the normalizing constant. Thus π(w) is smallest when w = id and, as θ → 1, π tends to the uniform distribution. These non-uniform distributions arise in statistical work as Mallows models. Background and references are in Section 2e.A standard Monte Carlo Markov chain algorithm for sampling from π is the Metropolis algorithm with a systematic scan. This algorithm cycles through the generators in order. If multiplying by the current generator increases length this multiplication is made. If the length decreases, then the multiplication is made with probability θ and omitted with probability 1 − θ. One scan uses s , in order. Define K(w, w ′ ) = the chance that a systematic scan started at w ends in w ′ .(1.2)Repeated scans of the algorithm are defined byIn Section 2c and 4a we show that this Markov chain has π as unique stationary distribution.The main results of this paper derive sharp results on rates of convergence for these walks. As an example of what our methods give, we show that order n scans are necessary and suffice to reach stationary on the symmetric group starting from the identity. More precisely, we prove

We show how the ribbon Hopf algebra structure on the Drinfel'd Jimbo quantum groups of Types A, B, C, and D can be used to derive formulas giving explicit realizations of the irreducible representations of the Iwahori Hecke algebras of type A and the Birman Wenzl algebras. We use this derivation to give explicit realizations of the irreducible representations of the Brauer algebras as well. The derivation is accomplished by way of a combination of techniques from operator algebras, quantum groups, and the theory of 3-manifold invariants. Although our applications are in the cases of the quantum groups of Types A, B, C, and D, most of the aspects of our approach apply in the general setting of ribbon Hopf algebras.

Groundwater is an important source for drinking water supply in hard rock terrain of Bundelkhand massif particularly in District Mahoba, Uttar Pradesh, India. An attempt has been made in this work to understand the suitability of groundwater for human consumption. The parameters like pH, electrical conductivity, total dissolved solids, alkalinity, total hardness, calcium, magnesium, sodium, potassium, bicarbonate, sulfate, chloride, fluoride, nitrate, copper, manganese, silver, zinc, iron and nickel were analysed to estimate the groundwater quality. The water quality index (WQI) has been applied to categorize the water quality viz: excellent, good, poor, etc. which is quite useful to infer the quality of water to the people and policy makers in the concerned area. The WQI in the study area ranges from 4.75 to 115.93. The overall WQI in the study area indicates that the groundwater is safe and potable except few localized pockets in Charkhari and Jaitpur Blocks. The Hill-Piper Trilinear diagram reveals that the groundwater of the study area falls under Na+-Cl−, mixed Ca2+-Mg2+-Cl− and Ca2+-$${\text{HCO}}_{3}^{ - }$$ HCO 3 - types. The granite-gneiss contains orthoclase feldspar and biotite minerals which after weathering yields bicarbonate and chloride rich groundwater. The correlation matrix has been created and analysed to observe their significant impetus on the assessment of groundwater quality. The current study suggests that the groundwater of the area under deteriorated water quality needs treatment before consumption and also to be protected from the perils of geogenic/anthropogenic contamination.

Brauer's centralizer algebras are finite dimensional algebras with a distinguished basis. Each Brauer centralizer algebra contains the group algebra of a symmetric group as a subalgebra and the distinguished basis of the Brauer algebra contains the permutations as a subset. In view of this containment it is desirable to generalize as many known facts concerning the group algebra of the symmetric group to the Brauer algebras as possible. This paper studies the irreducible characters of the Brauer algebras in view of the distinguished basis. In particular we define an analogue of conjugacy classes, and derive Frobenius formulas for the characters of the Brauer algebras. Using the Frobenius formulas we derive formulas for the irreducible character of the Brauer algebras in terms of the irreducible characters of the symmetric groups and give a combinatorial rule for computing these irreducible characters.Introduction. Classically, Frobenius [Fr] determined the characters of the symmetric group by exploiting the connection between the power symmetric functions and the Schur functions. Schur [Scl, Sc2] later showed that this connection arises from the fact that the general linear group and the symmetric group each generate the full centralizer of each other on tensor space, now referred to as the Schur-Weyl duality. In his landmark book [Wy], Weyl used this duality as the principal algebraic tool for studying the representations of the classical groups. In 1937 R. Brauer [Br] gave a nice basis for the centralizer algebra of the action of the orthogonal and symplectic groups on tensor space.In [Rl] we gave a formula for the characters of the Hecke algebra of type A in the same spirit as the original formula of Frobenius for the characters of the symmetric group. This formula was then used to derive a combinatorial rule for computing the characters of the

Dedicated to R. MacPherson on the occasion of his 60th birthday IntroductionTogether, Sections 2 and 5 of this paper form a self contained treatment of the theory of crystals and the path model. It is my hope that this will be useful to the many people who, over the years, have told me that they wished they understood crystals but have found the existing literature too daunting. One goal of the presentation here is to clarify the relationship between the general path model and the crystal operators of Lascoux and Schützenberger used in the type A case [LS]. More specifically, Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model which is designed for readers whose only background is the material in Section 2. These two sections can be read independently of Sections 3 and 4.Sections 3 and 4 give an exposition of the affine Hecke algebra and recent results regarding the combinatorics of spherical functions on p-adic groups (Hall-Littlewood polynomials) using only the material in Section 2 as background. The q-analogue of the theory of crystals developed in Section 4 specializes to the path model version of the "classical" crystal theory at q −1 = 0. This specialization property is "to be expected" since Macdonald's formula for the spherical function specializes to the Weyl character formula at q −1 = 0. It is my hope that the presentation of the results will illustrate the close connection between the affine Hecke algebra, the path model, and the theory of crystals.The motivation for this paper is the following. The classical path model is a combinatorial tool for working with crystal bases of integrable representations of symmetrizable Kac-Moody algebras, a generalization of column strict Young tableaux. The same combinatorics can be used to study the equivariant K-theory of the flag variety [PR] and this point provides a translation between the path model and the structure of the nil-affine Hecke algebra. Thus, conceptually, crystal bases = the path model = the nil affine Hecke algebra = the T -equivariant K-theory of G/B.The paper of Gaussent-Littelmann [GL] (see also the work of Kapovich-Millson in [KM]) indicated that there is a refined model which removes the 'nil' in this chain of equalities and models 1 the geometry of Kac-Moody groups over a local field. The result is the conceptual chain of equalities positively folded alcove walks = geodesics in the affine building = the affine Hecke algebra = MV cycles in the loop Grassmanian.The connection to the affine Hecke algebra and the approach to spherical functions for a p-adic group in [NR] was made concrete by C. Schwer [Sc] who told me that "the periodic Hecke module encodes the positively folded galleries".The main results of this paper are obtained by viewing the affine Hecke algebra in terms of a new construct, the alcove walk algebra. This combinatorics provides in the same way that the theory of crystal bases gives...

We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r, p, n) and the corresponding cyclotomic Hecke algebras H r,p,n , can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.

The purpose of this paper is to describe a general procedure for computing analogues of Young's seminormal representations of the symmetric groups. The method is to generalize the Jucys-Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras. The combinatorics of these elements allows one to compute irreducible representations explicitly and often very easily. In this paper we do these computations for Weyl groups and Iwahori-Hecke algebras of types A n , B n , D n , G 2 . Although these computations are in reach for types F 4 , E 6 , and E 7 , we shall, in view of the length of the current paper, postpone this to another work. IntroductionThis paper is an attempt to simplify, unify, review, and expand the theory of seminormal representations of the symmetric groups. It is my hope that the material in this paper will be clear enough and general enough to dispel some of the mystique of this subject. I would suggest that the reader begin with section 3, the symmetric group case, and go back and pick up the generalities from sections 1 and 2 as they are needed. This will make In such a subject, where the credit for the discoveries has been traditionally complicated and mixed up, it is di cult to claim credit for much of anything. I believe that my general approach to seminormal representations of Weyl groups and Iwahori-Hecke algebras given in sections 1 and 2 of this paper is new, at least nobody explained it to me before. If this was the structure motivating the work of Hoefsmit H], then he certainly didn't give any hint of it, he produced the seminormal representations out of thin air and then proved that they were correct. I believe that my realization that the Jucys-Murphy elements are coming from the very natural central elements in (2.1) and Proposition (2.4) is new. I also think that before this paper nobody has realized that there is a simple concrete connection (Proposition (2.8)) between Jucys-Murphy type elements in Iwahori-Hecke algebras and Jucys-Murphy elements in group algebras of Weyl groups (even for the special cases). I know that the analogues of the Jucys-Murphy elements in Weyl groups of types B and D will be new to some of the experts and known to others. These Jucys-Murphy elements for types B and D are not new, similar elements appear in the paper of Cherednik Ch], but I was not able to recognize them there until they were pointed out to me by M.L. Nazarov. I extend my thanks to him for this. Some people were asking me for JucysMurphy elements in type G 2 as late as June of 1995. In July 1995 I was told that it was unknown how to quantize the elements of Cherednik, i.e. nd analogues of them in the Iwahori-Hecke algebras of types B and D. Of course, this has been done already in 1974, by Hoefsmit.The idea behind my approach to seminormal representations is as old as representation theory, it is the same method that gives us Gelfand-Tsetlin bases, and that is often used to nd the representations of centralizer algebras and ope...

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