Representations of the q-rook monoid

Halverson, Tom

doi:10.1016/j.jalgebra.2003.11.002

Cited by 39 publications

(68 citation statements)

References 15 publications

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“…Part (a) was proved in a different way by Solomon [So4] and part (b) is the result of Halverson [Ha,Theorem 3.2] which was the catalyst for the results of this paper.…”

Section: Hecke Algebrasmentioning

confidence: 80%

q-Rook Monoid Algebras, Hecke Algebras, and Schur–Weyl Duality

Halverson

Ram

2004

Journal of Mathematical Sciences

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The rook monoid R k is the monoid of k × k matrices with entries from {0, 1} and at most one nonzero entry in each row and column. Recently, the representation theory of its "IwahoriHecke" algebra R k (q), called the q-rook monoid algebra, has been analyzed. In particular, a Schur-Weyl type duality on tensor space was found for the q-rook monoid algebra and its irreducible representations were given explicit combinatorial constructions. In this paper we show that, in fact, the q-rook monoid algebra is a quotient of the affine Hecke algebra of type A. With this knowledge in hand, we show that the recent results on the q-rook monoid algebras actually come from known results about the affine Hecke algebra. In particular (a) The recent combinatorial construction of the irreducible representations of R k (q) byHalverson . Though these results show that the representation theory of the q-rook monoid algebra is "just" a piece of the representation theory of the affine Hecke algebra, this was not at all obvious at the outset. It was only on the analysis of the recent results in [So4] and [Ha] that the similarity to affine Hecke algebra theory was noticed. This observation then led us to search for and establish a concrete connection between these algebras.The q-rook monoid algebra was first studied in its q = 1 version in the 1950's by Munn [Mu1-2]. Solomon [So1] discovered the general q-version of the algebra as a Hecke algebra (double coset algebra) for the finite algebraic monoid M n (F q ) of n × n matrices over a finite field with q elements, with respect to the "Borel subgroup" B of invertible upper triangular matrices. Later Solomon [So2] found a Schur-Weyl duality for R k (1) in which R k (1) acts as the centralizer algebra for the action of the general linear group GL n (C) on V ⊗k where V = L(ε 1 ) ⊕ L(0) is the direct sum of the "fundamental" n-dimensional representation and the trivial module L(0) for GL n (C). Then Solomon [So3,4] gave a presentation of R k (q) by generators and relations and defined an action of R k (q) on tensor space.Halverson [Ha] found a new presentation of R k (q) and used it to show that Solomon's action of R k (q) on tensor space extends the Schur-Weyl duality so that R k (q) is the centralizer of the quantum general linear group U q gl(n) on V ⊗k where now V = L(ε 1 ) ⊕ L(0) is the direct sum of the "fundamental" and the trivial module for U q gl(n). Halverson also exploited his new presentation to construct, combinatorially, all the irreducible representations of R k (q) when R k (q) is semisimple.The main results of this paper are the following:(a) We find yet another presentation (1.6) of R k (q) by generators and relations. (b) Our new presentation shows thatwhere H k (0, 1; q) is the Iwahori-Hecke algebra of type B k with parameters specialized to 0 and 1, and I is the ideal generated by the minimal ideal of H 2 (0, 1; q) corresponding to the pair of partitions λ = ((1 2 ), ∅).(c) We show that the irreducible representations of R k (q) found in [Ha] come from the constructi...

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“…Part (a) was proved in a different way by Solomon [So4] and part (b) is the result of Halverson [Ha,Theorem 3.2] which was the catalyst for the results of this paper.…”

Section: Hecke Algebrasmentioning

confidence: 80%

q-Rook Monoid Algebras, Hecke Algebras, and Schur–Weyl Duality

Halverson

Ram

2004

Journal of Mathematical Sciences

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“…Just as an example we mention the highly interesting paper [34], where representations of the q-rook monoid are given. A q-rook monoid is a q-algebra with certain q-deformed commutation relations.…”

Section: Introductionmentioning

confidence: 99%

The different tongues of q-calculus

Ernst¹

2008

Proc. Estonian Acad. Sci.

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Abstract. In this review paper we summarize the various dialects of q-calculus: quantum calculus, time scales, and partitions. The close connection between Γ q (x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notation will be illustrated by the (q-)Euler reflection formula, q-Appell functions, CarlitzAlSalam polynomials, and the so-called q-addition. We conclude with some short biographies about famous scientists in q-calculus.

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“…The Clausen algorithm for FFT on the symmetric group S n requires a set of inequivalent, irreducible, chain-adapted representations relative to the chain S n > S n−1 > ··· > S 1 = {e}, where e is the identity of S n and S k is the subgroup of S n consisting of permutations which fix k + 1 through n. Two such sets of representations are provided by the Young seminormal (62) and orthogonal forms (77). (for generalizations to seminormal representations of Iwahori-Hecke algebras see (35) and (62)). A partition of a nonnegative integer k is a weakly decreasing sequence λ of nonnegative integers whose sum is k. Two partitions are equal if they only differ in number of zeros they contain.…”

Section: Fast Fourier Transformmentioning

confidence: 99%

“…Moreover, put τ i = σ n σ n−1 ...σ i+1 , for each i, and let R k be the subsemigroup of R n consisting of those partial permutations σ with σ(j)=j, for j > k. Then Halverson has characterizedR n using the semigroup chain R n > R n−1 > ··· > R 1 , similar to the Young seminormal representations of S n (35). Considering the transversal decomposition of R n into cosets of R n−1 , one should note that two distinct left cosets of R n−1 do not necessarily have the same cardinality.…”

Section: Fast Fourier Transform On Rook Monoidsmentioning

confidence: 99%