@article{putcha_2017, title={Idempotent conjugacy in monoids}, volume={95}, ISSN={["1432-2137"]}, DOI={10.1007/s00233-016-9818-5}, number={2}, journal={SEMIGROUP FORUM}, author={Putcha, Mohan S.}, year={2017}, month={Oct}, pages={366–378} }
 @article{putcha_2014, title={Centralizer of an idempotent in a reductive monoid}, volume={26}, ISSN={["1435-5337"]}, DOI={10.1515/form.2011.163}, abstractNote={Abstract Let}, number={2}, journal={FORUM MATHEMATICUM}, author={Putcha, Mohan S.}, year={2014}, month={Mar}, pages={323–335} }
 @article{putcha_2011, title={Canonical semigroups}, volume={83}, ISSN={["0037-1912"]}, DOI={10.1007/s00233-011-9307-9}, number={1}, journal={SEMIGROUP FORUM}, author={Putcha, Mohan S.}, year={2011}, month={Aug}, pages={65–74} }
 @article{putcha_2011, title={ROOT SEMIGROUPS IN REDUCTIVE MONOIDS}, volume={21}, ISSN={["0218-1967"]}, DOI={10.1142/s0218196711006261}, abstractNote={ It is well known that in a reductive group, the Borel subgroup is a product of the maximal torus and the one-dimensional positive root subgroups. The purpose of this paper is to find an analog of this result for reductive monoids. Via a study of reductive monoids of semisimple rank 1, we introduce the concept of root semigroups. By analyzing the associated root elements in the Renner monoid, we show that the closure of the Borel subgroup is generated by the maximal torus and positive root semigroups. Along the way we generalize the Jordan decomposition of algebraic groups to reductive monoids. }, number={3}, journal={INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION}, author={Putcha, Mohan S.}, year={2011}, month={May}, pages={433–448} }
 @article{putcha_2008, title={Nilpotent variety of a reductive monoid}, volume={27}, ISSN={["0925-9899"]}, DOI={10.1007/s10801-007-0087-y}, abstractNote={In this paper we study the variety M nil of nilpotent elements of a reductive monoid M. In general this variety has a completely different structure than the variety G uni of unipotent elements of the unit group G of M. When M has a unique non-trivial minimal or maximal G×G-orbit, we find a precise description of the irreducible components of M nil via the combinatorics of the Renner monoid of M and the Weyl group of G. In particular for a semisimple monoid M, we find necessary and sufficient conditions for the variety M nil to be irreducible.}, number={3}, journal={JOURNAL OF ALGEBRAIC COMBINATORICS}, author={Putcha, Mohan S.}, year={2008}, month={May}, pages={275–292} }
 @article{putcha_2007, title={Monoids and cuspidal group characters}, volume={75}, ISSN={["1432-2137"]}, DOI={10.1007/s00233-007-0715-9}, number={3}, journal={SEMIGROUP FORUM}, author={Putcha, Mohan S.}, year={2007}, pages={544–554} }
 @article{putcha_2006, title={Idempotent-conjugate monoids with nilpotent unit groups}, volume={72}, ISSN={["1432-2137"]}, DOI={10.1007/s00233-005-0560-7}, number={2}, journal={SEMIGROUP FORUM}, author={Putcha, MS}, year={2006}, pages={329–336} }
 @article{putcha_2006, title={Parabolic monoids I. Structure}, volume={16}, ISSN={["0218-1967"]}, DOI={10.1142/S0218196706003414}, abstractNote={ We determine the closure of a parabolic subgroup of a reductive group in a reductive monoid. This allows us to define parabolic submonoids of a finite monoid of Lie type. These are analogues of the monoid of block upper triangular matrices. We determine the structure of [Formula: see text]-class of a finite parabolic monoid and show that such a monoid is generated by its unit group and diagonal idempotents. }, number={6}, journal={INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION}, author={Putcha, Mohan S.}, year={2006}, month={Dec}, pages={1109–1129} }
 @article{putcha_2005, title={Conjugacy decomposition of reductive monoids}, volume={250}, ISSN={["0025-5874"]}, DOI={10.1007/s00209-005-0779-8}, number={4}, journal={MATHEMATISCHE ZEITSCHRIFT}, author={Putcha, MS}, year={2005}, month={Aug}, pages={841–853} }
 @article{putcha_2004, title={Brauer characters of finite monoids}, volume={7}, ISSN={["1386-923X"]}, DOI={10.1023/B:ALGE.0000019387.07748.9b}, number={1}, journal={ALGEBRAS AND REPRESENTATION THEORY}, author={Putcha, MS}, year={2004}, month={Mar}, pages={59–66} }
 @article{putcha_2004, title={Bruhat-Chevalley order in reductive monoids}, volume={20}, ISSN={["1572-9192"]}, DOI={10.1023/B:JACO.0000047291.42015.a6}, abstractNote={Let M be a reductive monoid with unit group G. Let Λ denote the idempotent cross-section of the G × G-orbits on M. If W is the Weyl group of G and e, f ∈ Λ with e ≤ f, we introduce a projection map from WeW to WfW. We use these projection maps to obtain a new description of the Bruhat-Chevalley order on the Renner monoid of M. For the canonical compactification X of a semisimple group G 0 with Borel subgroup B 0 of G 0, we show that the poset of B 0 × B 0-orbits of X (with respect to Zariski closure inclusion) is Eulerian.}, number={1}, journal={JOURNAL OF ALGEBRAIC COMBINATORICS}, author={Putcha, MS}, year={2004}, month={Jul}, pages={33–53} }
 @article{putcha_2004, title={Mobius function on cross-section lattices}, volume={106}, ISSN={["0097-3165"]}, DOI={10.1016/j.jcta.2004.03.001}, abstractNote={Let M be an irreducible algebraic monoid with a reductive unit group G . Then there is an idempotent cross-section Λ of G × G -orbits that preserves the Zariski closure ordering. The purpose of this paper is to compute the Möbius function on the cross-section lattice Λ . This is accomplished by analyzing an associated boolean family of face lattices of polytopes and then solving a resulting system of linear equations.}, number={2}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Putcha, MS}, year={2004}, month={May}, pages={287–297} }
 @article{putcha_2002, title={Big cells and LU factorization in reductive monoids}, volume={130}, ISSN={["0002-9939"]}, DOI={10.1090/S0002-9939-02-06515-2}, abstractNote={It is well known that an invertible matrix admits a factorization as a product of a lower triangular matrix L and an upper triangular matrix U if and only if all the principal minors of the matrix are non-zero. The corresponding problem for singular matrices is much more subtle. We study this problem in the general setting of a reductive monoid and obtain a solution in terms of the Bruhat-Chevalley order. In the process we obtain a decomposition of the big cell B-B of a reductive monoid, where B and B- are opposite Borel subgroups of the unit group.}, number={12}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Putcha, MS}, year={2002}, pages={3507–3513} }
 @article{putcha_2002, title={Shellability in reductive monoids}, volume={354}, ISSN={["0002-9947"]}, DOI={10.1090/S0002-9947-01-02806-9}, abstractNote={The purpose of this paper is to extend to monoids the work of Bjorner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let M be a reductive monoid with unit group G, Borel subgroup B and Weyl group W. We study the partially ordered set of B x B-orbits (with respect to Zariski closure inclusion) within a G x G-orbit of M. This is the same as studying a W x W-orbit in the Renner monoid R. Such an orbit is the retract of a 'universal orbit', which is shown to be lexicograhically shellable in the sense of Bjorner and Wachs.}, number={1}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Putcha, MS}, year={2002}, pages={413–426} }
 @article{putcha_2001, title={Monoids with idempotent cross-sections}, volume={11}, ISSN={["0218-1967"]}, DOI={10.1142/S0218196701000632}, abstractNote={ In this paper we consider monoids M with an idempotent cross-section Λ of regular [Formula: see text]-classes such that for all idempotents e, f of M with e≥f, there exists x in the unit group such that x-1ex, x-1fx∈Λ. Such monoids arise frequently in connection with group representations. We show that Λ is then necessarily a semilattice and that the idempotent structure of M can for the most part, be determined within the unit group. }, number={4}, journal={INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION}, author={Putcha, MS}, year={2001}, month={Aug}, pages={457–466} }
 @article{putcha_2001, title={Reciprocity in character theory of finite semigroups}, volume={163}, ISSN={["0022-4049"]}, DOI={10.1016/S0022-4049(00)00162-6}, abstractNote={This paper concerns the complex characters of finite semigroups. We begin by reducing the general problem to the study of local monoids M = G ∪ J ∪{0}, where G is the unit group and J is a regular J -class. Let H be a maximal subgroup of J . For a C H -module U with character θ , let Ũ be the associated C M -module and let θ ̃ be the character of the C G -module Ũ . For a C G -module V with character ψ , we construct a natural C H -module V ∼ with character ψ ∼ . We prove the reciprocity theorem: 〈 θ ̃ ,ψ〉 G =〈θ, ψ ∼ 〉 H . As a consequence we find a characterization of the J -cuspidal characters of G . We go on to interpret our results for groups, introducing the concept of double parabolic induction.}, number={3}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Putcha, MS}, year={2001}, month={Oct}, pages={339–351} }
 @article{putcha_2000, title={Semigroups and weights for group representations}, volume={128}, ISSN={["0002-9939"]}, DOI={10.1090/S0002-9939-00-05464-2}, abstractNote={Let G be a finite group. Consider a pair χ = (χ+, χ−) of linear characters of subgroups P,P− of G with χ+ and χ− agreeing on P ∩ P−. Naturally associated with χ is a finite monoid Mχ. Semigroup representation theory then yields a representation θ of G. If θ is irreducible, we say that χ is a weight for θ. When the underlying field is the field of complex numbers, we obtain a formula for the character of θ in terms of χ+ and χ−. We go on to construct weights for some familiar group representations.}, number={10}, journal={PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Putcha, MS}, year={2000}, pages={2835–2842} }
 @article{okninski_putcha_2000, title={Subspace semigroups}, volume={233}, ISSN={["0021-8693"]}, DOI={10.1006/jabr.2000.8417}, abstractNote={For a finite dimensional algebra A over an infinite field K, the subspace semigroup S(A) consists of all subspaces of A with operation V ∗ W = linK(VW). We describe the structure of S(A), showing in particular that, similarly to any algebraic linear semigroup, S(A) is strongly π-regular; we describe its regular elements and regular T-classes. As the key intermediate step, for an arbitrary connected algebraic monoid M, we study the semigroup C(M) consisting of all irreducible closed subsets with operation X · Y = XY, and we transfer the information to S(A) via the natural onto homomorphism C(A) → S(A). With potential applications in mind, our primary focus in this paper is on the case of the full matrix algebra A = Mn(K).}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Okninski, J and Putcha, MS}, year={2000}, month={Nov}, pages={87–104} }
 @article{putcha_1999, title={Hecke algebras and semisimplicity of monoid algebras}, volume={218}, ISSN={["0021-8693"]}, DOI={10.1006/jabr.1999.7868}, abstractNote={Abstract We begin by constructing Hecke algebras for arbitrary finite regular monoids M. We then show that the semisimplicity of the complex monoid algebra C M is equivalent to the semisimplicity of the associated Hecke algebras and a condition on induced group characters. We apply these results to finite Lie-type monoids M. We show that the monoid algebra FM over a field F is semisimple if and only if the characteristic of F does not divide the order of the unit group G. This is accomplished by developing formulas for the unities of C J, J a J -class of M. The unity is explicitly given when G is a simply connected Chevalley group and J is associated with a Borel subgroup of G.}, number={2}, journal={JOURNAL OF ALGEBRA}, author={Putcha, MS}, year={1999}, month={Aug}, pages={488–508} }
 @article{putcha_1999, title={Invariant algebra and cuspidal representations of finite monoids}, volume={212}, ISSN={["0021-8693"]}, DOI={10.1006/jabr.1998.7634}, abstractNote={Abstract Motivated by the theories of Hecke algebras and Schur algebras, we consider in this paper the algebra C MGofG-invariants of a finite monoidMwith unit groupG. IfMis a regular “balanced” monoid, we show that C MGis a quasi-hereditary algebra. In such a case, we find the blocks of C MGto be the “sections” of the blocks of C M. We go on to develop a theory of cuspidal representations for balanced monoids. We then apply our results to the full transformation semigroup and the multiplicative monoid of triangular matrices over a finite field.}, number={2}, journal={JOURNAL OF ALGEBRA}, author={Putcha, MS}, year={1999}, month={Feb}, pages={721–737} }
 @article{li_putcha_1999, title={Types of reductive monoids}, volume={221}, DOI={10.1006/jabr.1999.7946}, abstractNote={Abstract Let M be a reductive monoid with a reductive unit group G. Clearly there is a natural G × G action on M. The orbits are the J -classes (in the sense of semigroup theory) and form a finite lattice. The general problem of finding the lattice remains open. In this paper we study a new class of reductive monoids constructed by multilined closure. We obtain a general theorem to determine the lattices of these monoids. We find that the ( J , σ)-irreducible monoids of Suzuki type and Ree type belong to this new class. Using the general theorem we then list all the lattices and type maps of the ( J , σ)-irreducible monoids of Suzuki type and Ree type.}, number={1}, journal={Journal of Algebra}, author={Li, Zhilin and Putcha, M.}, year={1999}, pages={102–116} }
 @article{putcha_1998, title={Complex representations of finite monoids II. Highest weight categories and quivers}, volume={205}, ISSN={["0021-8693"]}, DOI={10.1006/jabr.1997.7395}, abstractNote={Abstract In this paper we continue our study of complex representations of finite monoids. We begin by showing that the complex algebra of a finite regular monoid is a quasi-hereditary algebra and we identify the standard and costandard modules. We define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules. We use our results to determine the blocks of the complex algebra of the full transformation semigroup. We show that there are only two blocks when the degree ≠ 3. We also show that when the degree ≥ 5, the complex algebra of the full transformation semigroup is not of finite representation type, answering negatively a conjecture of Ponizovskii.}, number={1}, journal={JOURNAL OF ALGEBRA}, author={Putcha, MS}, year={1998}, month={Jul}, pages={53–76} }
 @article{putcha_1998, title={Conjugacy classes and nilpotent variety of a reductive monoid}, volume={50}, ISSN={["0008-414X"]}, DOI={10.4153/CJM-1998-044-7}, abstractNote={Abstract}, number={4}, journal={CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES}, author={Putcha, MS}, year={1998}, month={Aug}, pages={829–844} }
 @article{putcha_1998, title={Monoid deformations and group representations}, volume={132}, ISSN={["0022-4049"]}, DOI={10.1016/S0022-4049(97)00125-4}, abstractNote={The purpose of this paper is to introduce the concept of monoid deformations in connection with group representations. The underlying philosophy for finite reductive monoids M is that while M is contained in a modular representation of the unit group G, a deformation M(q) is contained in a complex representation of G. This is worked out in detail in the case of the Steinberg representation.}, number={2}, journal={JOURNAL OF PURE AND APPLIED ALGEBRA}, author={Putcha, MS}, year={1998}, month={Nov}, pages={159–178} }
 @article{pennell_putcha_renner_1997, title={Analogue of the Bruhat-Chevalley order for reductive monoids}, volume={196}, ISSN={["0021-8693"]}, DOI={10.1006/jabr.1997.7111}, abstractNote={The purpose of this paper is to describe the Adherence Order of w x B = B-orbits on a reductive algebraic monoid. By the results of 10 there is already a perfect analogue of the Bruhat decomposition for reductive w x monoids. Indeed, by 10, Corollary 5.8 , if M is reductive with unit group G and maximal torus T : G with Borel subgroup B = T , then the set of two sided B-orbits, B R MrB is canonically identified with a certain finite Ž . inverse monoid R. In fact, R s N T rT , the orbit monoid of the Zariski G Ž . closure of N T in M. By definition, for s , t g R we define the AdherG ence Order by s F t if Bs B : Bt B, where Bt B denotes the Zariski closure in M of Bt B.}, number={2}, journal={JOURNAL OF ALGEBRA}, author={Pennell, EA and Putcha, MS and Renner, LE}, year={1997}, month={Oct}, pages={339–368} }
 @article{putcha_1997, title={Monoid Hecke algebras}, volume={349}, ISSN={["0002-9947"]}, DOI={10.1090/S0002-9947-97-01823-0}, abstractNote={This paper concerns the monoid Hecke algebras H introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group W . We use this to: 1. find a description of the irreducible representations of H, 2. find an explicit isomorphism between H and the monoid algebra of the Renner monoid R, 3. extend the Kazhdan-Lusztig involution and basis to H, and 4. prove, for a W ×W orbit of R, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials. Introduction A monoid analogue of the Iwahori-Hecke algebra [11] was obtained by Solomon [27]–[29]. In an earlier paper [19] the author studied Solomon’s monoid Hecke algebras by studying the associated orbit algebras. These orbit algebras arise from the two-sided action of the Weyl group W on the Renner monoid R. In particular, the coefficients of the unity of the empty level orbit algebra were shown to be Rx,y, where Rx,y are polynomials introduced by Kazhdan and Lusztig [12]. The other orbit algebras were also shown to have unities, but their coefficients were only implicitly given. In this paper we give an explicit formula for the unities of all the orbit algebras, thereby obtaining a description of the irreducible representations of monoid Hecke algebras. We also obtain an explicit, but very complicated, isomorphism between the monoid Hecke algebra and the monoid algebra of R, solving a problem posed by Solomon [28]. We go on to extend to the monoid Hecke algebra the Kazhdan-Lusztig involution and basis for the (group) Iwahori-Hecke algebra. This then immediately yields polynomials Pθ,σ for θ, σ in the same W ×W orbit of R, partially solving a problem posed by Renner [26]. These polynomials are still mysterious; however, in the simplest case they are products of relative KazhdanLusztig polynomials introduced by Deodhar [6]. 1. Reductive monoids and monoids of Lie type Consider the general linear group G = GLn(F ) over an algebraically closed field F . It is the unit group of the multiplicative monoidM = Mn(F ) of all n×nmatrices over F . This monoid has the following structure. The diagonal idempotents form a Boolean lattice with respect to the natural order of idempotents: f ≤ e if ef = fe = f. Received by the editors December 3, 1993. 1991 Mathematics Subject Classification. Primary 20G40, 20G05, 20M30. Research partially supported by NSF Grant DMS9200077. c ©1997 American Mathematical Society}, number={9}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Putcha, MS}, year={1997}, month={Sep}, pages={3517–3534} }