scholarly journals Explicit Pieri Inclusions

10.37236/9216 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Markus Hunziker ◽  
John A. Miller ◽  
Mark Sepanski
Keyword(s):  
Time Complexity ◽  
General Linear Group ◽  
Free Resolutions ◽  
Exponential Time ◽  
Exterior Power ◽  
Bounded Number ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
Multiplicity Free ◽  
Free Decomposition

By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents  are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fløystad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups. In this paper, we give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.

scholarly journals On the block structure of the quantum ℛ-matrix in the three-strand braids

2018 ◽  
Vol 33 (17) ◽  
pp. 1850105 ◽  
Author(s):  
L. Bishler ◽  
An. Morozov ◽  
Sh. Shakirov ◽  
A. Sleptsov
Keyword(s):  
Block Structure ◽  
Representation Formula ◽  
Building Blocks ◽  
Essential Condition ◽  
The Road ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation ◽  
On The Road ◽  
Homfly Polynomials ◽  
Block Diagonal

Quantum [Formula: see text]-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation [Formula: see text] of [Formula: see text] associated with each strand, one needs two matrices: [Formula: see text] and [Formula: see text]. They are related by the Racah matrices [Formula: see text]. Since we can always choose the basis so that [Formula: see text] is diagonal, the problem is reduced to evaluation of [Formula: see text]-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that [Formula: see text]-matrices could be transformed into a block-diagonal ones by the rotation in the sectors of coinciding eigenvalues. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of [Formula: see text] matrix. In this case in order to get a block-diagonal matrix, one should rotate the [Formula: see text] defined by the Racah matrix in the accidental sector by the angle exactly [Formula: see text].

REDUCING SIMPLE GRAMMARS: EXPONENTIAL AGAINST HIGHLY-POLYNOMIAL TIME IN PRACTICE

2007 ◽  
Vol 18 (04) ◽  
pp. 715-725
Author(s):  
CÉDRIC BASTIEN ◽  
JUREK CZYZOWICZ ◽  
WOJCIECH FRACZAK ◽  
WOJCIECH RYTTER
Keyword(s):  
Polynomial Time ◽  
Experimental Analysis ◽  
Time Complexity ◽  
Packet Classification ◽  
State Machines ◽  
Worst Case ◽  
Exponential Time ◽  
Push Down

Simple grammar reduction is an important component in the implementation of Concatenation State Machines (a hardware version of stateless push-down automata designed for wire-speed network packet classification). We present a comparison and experimental analysis of the best-known algorithms for grammar reduction. There are two approaches to this problem: one processing compressed strings without decompression and another one which processes strings explicitly. It turns out that the second approach is more efficient in the considered practical scenario despite having worst-case exponential time complexity (while the first one is polynomial). The study has been conducted in the context of network packet classification, where simple grammars are used for representing the classification policies.

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

On Degenerations of Modules With Nondirecting Indecomposable Summands

1996 ◽  
Vol 48 (5) ◽  
pp. 1091-1120 ◽  
Author(s):  
A. Skowroński ◽  
G. Zwara
Keyword(s):  
Natural Number ◽  
Partial Order ◽  
General Linear Group ◽  
Partial Orders ◽  
Connected Components ◽  
Closed Field ◽  
Affine Variety ◽  
Indecomposable Modules ◽  
Isomorphism Classes ◽  
Finite Dimensional

AbstractLet A be a finite dimensional associative K-algebra with an identity over an algebraically closed field K, d a natural number, and modA(d) the affine variety of d-dimensional A-modules. The general linear group Gld(K) acts on modA(d) by conjugation, and the orbits correspond to the isomorphism classes of d-dimensional modules. For M and N in modA(d), N is called a degeneration of M, if TV belongs to the closure of the orbit of M. This defines a partial order ≤deg on modA(d). There has been a work [1], [10], [11], [21] connecting ≤deg with other partial orders ≤ext and ≤deg on modA(d) defined in terms of extensions and homomorphisms. In particular, it is known that these partial orders coincide in the case A is representation-finite and its Auslander-Reiten quiver is directed. We study degenerations of modules from the additive categories given by connected components of the Auslander-Reiten quiver of A having oriented cycles. We show that the partial orders ≤ext, ≤deg and < coincide on modules from the additive categories of quasi-tubes [24], and describe minimal degenerations of such modules. Moreover, we show that M ≤degN does not imply M ≤ext N for some indecomposable modules M and N lying in coils in the sense of [4].

scholarly journals Exponential Time Complexity of the Permanent and the Tutte Polynomial

2014 ◽  
Vol 10 (4) ◽  
pp. 1-32 ◽  
Author(s):  
Holger Dell ◽  
Thore Husfeldt ◽  
Dániel Marx ◽  
Nina Taslaman ◽  
Martin Wahlén
Keyword(s):  
Time Complexity ◽  
Tutte Polynomial ◽  
Exponential Time

scholarly journals Multiplicity-free quotient tensor algebras

2001 ◽  
Vol 71 (2) ◽  
pp. 279-298
Author(s):  
G. E. Wall
Keyword(s):  
General Linear Group ◽  
Partial Solution ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Tensor Algebras ◽  
Completely Reducible ◽  
Multiplicity Free ◽  
Free Quotient

AbstractLet V be an infinite-dimensional vector space ovre a field of characteristic 0. It is well known that the tensor algebra T on V is a completely reducible module for the general linear group G on V. This paper is concerned with those quotient algebras A of T that are at the same time modules for G. A partial solution is given to the problem of determinig those A in which no irreducible constitutent has multiplicity greater thatn 1.

A SIMPLE ALGORITHM FOR COMPUTING THE GLOBAL CRYSTAL BASIS OF AN IRREDUCIBLE Uq(sln)-MODULE

2000 ◽  
Vol 10 (02) ◽  
pp. 191-208 ◽  
Author(s):  
BERNARD LECLERC ◽  
PHILIPPE TOFFIN
Keyword(s):  
Irreducible Representation ◽  
Simple Algorithm ◽  
Crystal Basis ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation

We describe a simple algorithm for computing Kashiwara's global crystal basis of a finite-dimensional irreducible representation of Uq(sln). Résumé: Nous décrivons un algorithme simple pour calculer la base cristalline globale de Kashiwara d'une représentation irréductible de dimension finie de Uq(sln).

scholarly journals Rationality of the generalized binomial coefficients for a multiplicity free action

2000 ◽  
Vol 68 (3) ◽  
pp. 387-410 ◽  
Author(s):  
Chal Benson ◽  
Gail Ratcliff
Keyword(s):  
Vector Space ◽  
Canonical Basis ◽  
Free Action ◽  
Homogeneous Component ◽  
Binomial Coefficients ◽  
Invariant Polynomials ◽  
Finite Dimensional ◽  
Multiplicity Free ◽  
Lie Subgroup ◽  
Generalized Binomial Coefficients

AbstractLetVbe a finite dimensional Hermitian vector space andKbe a compact Lie subgroup ofU(V) for which th representation ofKonC[V] is multiplicity free. One obtains a canonical basis {pα} for the spaceC[VR]kofK-invariant polynomials on VRand also a basis {q's. The polynomialpα's yields the homogeneous component of highest degree inqα. The coefficient that express theqα's in terms of thepβ's are the generalized binomial coeffficients of Yan. The main result in this paper shows tht these numbers are rational.

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