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The complex of Weyl module in free characteristic in the event of a partition (7,5,3)

  • Ghsoon Rabeea Raheem EMAIL logo and Haytham Razooki Hassan
Published/Copyright: January 18, 2024
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Open Engineering
From the journal Open Engineering Volume 14 Issue 1

Abstract

In this research, the diagrams and divided power, together with their Capelli’s identities, were employed to investigate “ the Complex of Lascoux ” in the situation of partition (7,5,3).

Keywords: divided power; complex of Lascoux; Capelli’s identity

1 Introduction

Let F stand for the “ free R_module ,” R be the ring containing commutative characteristics with 1, and D n F be the nth degree divided power [1]. In 2004, Buchsbaum modified the boundary method [2]. Boffi and Buchsbaum [3] then studied the complex of Lascoux (characteristic zero) in the partitions, which were (2,2,2) and (3,3,3), respectively. Using this modified approach and the Letter-Place Methods, Buchsbaum [4] and Sherouk and Niran [16] investigated the complex of Lascoux (characteristic zero) in the partition (4,4,3), technique in [4,17], obtaining characteristic zero. Consider the map

D p + k Ƒ ⨂ D q − k Ƒ → D p Ƒ ⨂ D q Ƒ . This map shows the polarization between places one and two, with place one and place two marked by D p + k Ƒ ⨂ D q − k Ƒ , respectively, and the map

∂ 32 ( k ) : : D p + k Ƒ ⨂ D q + k Ƒ ⨂ D r − k Ƒ → D p Ƒ ⨂ D q F ⨂ D r Ƒ is a location polarization that occurs from place two to place three [5,6,7,8,9,10]. We can request the identities in this situation as well.

The following equation must be used in a circumstance where ∂ 21 ∂ 32 is the result of looking in three different areas.

(1.1) ∂ 32 ( 1 ) ° ∂ 21 ( 1 ) − ∂ 21 ( 1 ) ° ∂ 32 ( 1 ) = ∂ 31 ( 1 ) .

According to previous studies [2,3,5], this is a typical Capelli’s identification

(1.2) ∂ 32 ( ᶄ ) ° ∂ 21 ( l ) = ∂ 32 ( ᶄ − a ) ° ∂ 31 ( a ) ; where ∂ ij ( 0 ) = I …

In general, the divided power of place polarizations meets the following identities in the case of ᶄ ≠ i:

(1.3) ∂ ij ( r ) ° ∂ j ᶄ ( s ) = ∑ a ≥ 0 ∂ j ᶄ ( s − a ) ° ∂ ij ( r − a ) ° ∂ i ᶄ ( a ) ,

(1.4) ∂ jk ( s ) ° ∂ ij ( r ) = ∑ a ≥ 0 ( − 1 ) a ∂ j ᶄ ( s − a ) ° ∂ ij ( r − a ) ° ∂ i ᶄ ( a ) ,

(1.5) ∂ 21 ( l ) ° ∂ 31 = ∂ 31 ° ∂ 21 ( ᶄ ) ,

(1.6) ∂ 32 ( ᶄ ) ° ∂ 31 = ∂ 31 ° ∂ 32 ( ᶄ ) .

Few researchers [11,12,13,14,15,16,17] studied the cases S L (2, U ) and P S L (2, U ), where U = 23, 29, 31, and 37. While Mohammed et al. [15] surveyed the case S L (2,38).

In this study, we investigate the Lascoux complex in the situation of the partition (7,5,3), Our objective is to gain a better knowledge of this complex and its attributes by employing various approaches such as homological, diagram, contracting homotopy, place polarization, and mapping cone methods.

The place polarization technique is a combinatorial approach for computing the Lascoux complex homology by counting particular sorts of pathways in a young tableau. Finally, the mapping cone method is a homological algebra methodology for computing the homology of a chain complex by contracting a cone over the complex. Specifically, Section 2 of this work describes the terms of Lascoux complex in the partition (7,5,3). Section 3 examines this complex as a graphic utilizing the split power of place polarization and its Capelli's identity .

2 The terms of Lascoux Complex in the case of partition (7,5,3)

In the case of partition (7,5,3), the terms of the Lascoux complex, as long as they match [1,4], can be committed to their locations according to the length of the permutation. We currently have the following matrix with the division λ = (7,5,3):

D 7 D 4 D 1 D 8 D 5 D 2 D 9 D 6 D 3 .

Then, the “ 'Lascous complex"' has the correspondence between its terms as follows:

D 7 ⨂ D 5 F ⨂ D 3 F ↔ identity ,

D 8 F ⨂ D 4 F ⨂ D 3 F ↔ (1 2),

D 7 F ⨂ D 6 F ⨂ D 2 F ↔ (2 3),

D 9 F ⨂ D 4 F ⨂ D 2 F ↔ (1 2 3),

D 8 F ⨂ D 6 F ⨂ D 1 F ↔ (1 3 2),

D 9 F ⨂ D 5 F ⨂ D 1 F ↔ (1 3).

So, the complex of Lascoux in the case of the partition ʎ = (7,5,3) has the form

3 The diagram of "the Complex of Lascoux" in the case of the partition (7,5,3)

Look at the following diagram:

So, if we define

n 1 : D 9 F ⨂ D 5 F ⨂ D 1 F → D 9 F ⨂ D 4 F ⨂ D 2 F as n 1 ( v ) = ∂ 32 ( v ) ; where v ∈ D 9 F ⨂ D 5 F ⨂ D 1 F ,

k 1 : D 9 F ⨂ D 5 F ⨂ D 1 F → D 8 F ⨂ D 6 F ⨂ D 1 F as k 1 ( v ) = ∂ 21 ( v ) ; where v ∈ D 9 F ⨂ D 5 F ⨂ D 1 F ,

and

k 2 : D 9 F ⨂ D 4 F ⨂ D 2 F → D 7 F ⨂ D 6 F ⨂ D 2 F as k 2 ( v ) = ∂ 21 ( 2 ) ( v ) ; where v ∈ D 9 F ⨂ D 4 F ⨂ D 2 F .

We must now define the following map in order to make the diagram L commutative:

m 1 : D 8 F ⨂ D 6 F ⨂ D 1 F → D 7 F ⨂ D 6 F ⨂ D 2 F ,

which implies that

m 1 ° k 1 = n 1 ° k 2 ,

m 1 ° ∂ 21 = ∂ 21 ( 2 ) ° ∂ 32 .

Now we use Capelli’s identities form

∂ 21 ( 2 ) ° ∂ 32 = n 1 ° ∂ 21 ,

∂ 21 ( 2 ) ° ∂ 32 = ∂ 32 ° ∂ 21 ( 2 ) − ∂ 31 ° ∂ 21 ,

= 1 2 ∂ 32 ° ∂ 21 − ∂ 31 ° ∂ 21 .

Thus, m 1 = 1 2 ∂ 32 ° ∂ 21 − ∂ 31 .

On the other hand, if we define

m 2 : D 7 F ⨂ D 6 F ⨂ D 2 F → D 7 F ⨂ D 5 F ⨂ D 3 F ,

m 2 ( v ) = ∂ 32 ( v ) ; where v ∈ D 7 F ⨂ D 6 F ⨂ D 2 F ,

and

k 3 : D 8 F ⨂ D 4 F ⨂ D 3 F → D 7 F ⨂ D 5 F ⨂ D 3 F ,

as k 3 (v) = ∂ 21 (v); where v ∈ D 8 F ⨂ D 4 F ⨂ D 3 F

To have the diagram R commute, we must define n 2 .

n 2 : D 9 F ⨂ D 4 F ⨂ D 2 F → D 8 F ⨂ D 4 F ⨂ D 3 F ,

such that

m 2 ° k 2 = k 3 ° n 2 i.e. ∂ 32 ° ∂ 21 ( 2 ) = ∂ 21 ° n 2 .

Again, by using Capelli identities we obtain

∂ 32 ° ∂ 21 ( 2 ) = ∂ 21 ( 2 ) ° ∂ 32 + ∂ 21 ° ∂ 31 = ∂ 21 ° 1 2 ∂ 21 ° ∂ 32 + ∂ 31 ,

Then, n 2 = 1 2 ∂ 21 ° ∂ 32 + ∂ 31 .

Now consider the following diagram:

Define X: D 8 F ⨂ D 6 F ⨂ D 1 F → D 8 F ⨂ D 4 F ⨂ D 3 F ,

by X ( v ) = ∂ 32 ( 2 ) ; where v ∈ D 8 F ⨂ D 6 F ⨂ D 1 F .

Proposition 3.2.1

The graph W is commutative.

Proof

In order to demonstrate that W is commutative, we must first demonstrate n 2 ° n 1 = X ° K 1 .

n 2 ° n 1 = 1 2 ∂ 21 ° ∂ 32 + ∂ 31 ∂ 32 = ∂ 21 ° ∂ 32 ( 2 ) + ∂ 32 ° ∂ 31

n 2 ° n 1 = ∂ 32 ( 2 ) ° ∂ 21 − ∂ 32 ° ∂ 31 + ∂ 32 ° ∂ 31 = ∂ 32 ( 2 ) ° ∂ 21 = X ° k 1 .

Proposition 3.2.2

The graph G is commutative.

Proof

m 2 ° m 1 = ∂ 32 ° 1 2 ∂ 32 ° ∂ 21 − ∂ 31 = ∂ 32 ( 2 ) ° ∂ 21 − ∂ 32 ° ∂ 31 = ∂ 21 ° ∂ 32 ( 2 ) + ∂ 32 ° ∂ 31 − ∂ 32 ° ∂ 31 = ∂ 21 ° ∂ 32 ( 2 ) = k 3 ° X .

Finally, we may define the maps σ 1 , σ 2 , and σ 3 by using the mapping Cone.

D 9 F ⨂ D 4 F ⨂ D 2 F σ 3 : D 9 F ⨂ D 5 F ⨂ D 1 F → ⨁ D 8 F ⨂ D 6 F ⨂ D 1 F ,

D 9 F ⨂ D 4 F ⨂ D 2 F D 8 F ⨂ D 4 F ⨂ D 3 F σ 2 : ⨁ → ⨁ D 8 F ⨂ D 6 F ⨂ D 1 F D 7 F ⨂ D 6 F ⨂ D 2 F ,

and

D 8 F ⨂ D 4 F ⨂ D 3 F σ 1 : ⨁ → D 7 F ⨂ D 5 F ⨂ D 3 F D 7 F ⨂ D 6 F ⨂ D 2 F ,

by

* σ 3 ( x ) = ( n 1 ( x ) , k 1 ( x ) ) ; ∀ x ∈ D 9 F ⨂ D 5 F ⨂ D 1 F = ( ∂ 32 ( x ) , ∂ 21 ( x ) ) ,

D 9 F ⨂ D 4 F ⨂ D 2 F * σ 2 ( x 1 , x 2 ) = ( n 2 ( x 1 ) – X ( x 2 ) , k 1 ( x 2 ) ⨁ − k 2 ( x 1 ) ) ; ∀ ( x 1 , x 2 ) ∈ D 8 F ⨂ D 6 F ⨂ D 1 F ,

D 8 F ⨂ D 4 F ⨂ D 3 F * σ 1 ( x 1 , x 2 ) = ( k 3 ( x 1 ) + m 2 ( x 2 ) ) ; ∀ ( x 1 , x 2 ) ∈ ⨁ D 7 F ⨂ D 6 F ⨂ D 2 F .

Proposition 3.2.3

D 9 F ⨂ D 4 F ⨂ D 2 F D 8 F ⨂ D 4 F ⨂ D 3 F 0 → D 9 F ⨂ D 5 F ⨂ D 1 F → σ 3 ⨁ → σ 2 ⨁ → σ 1 D 7 F ⨂ D 5 F ⨂ D 3 F D 8 F ⨂ D 6 F ⨂ D 1 F D 7 F ⨂ D 6 F ⨂ D 2 F

is complex.

Proof

Since ∂ 32 ( 2 ) and ∂ 21 are injective as defined in [2], then we get σ 3 .

Now

σ 2 ° σ 3 = σ 2 ( n 1 ( ӽ ) , k 1 ( ӽ ) ) = σ 2 ( ∂ 32 ( ӽ ) , ∂ 21 ( ӽ ) ) = ( n 2 ( ∂ 32 ( ӽ ) ) – X ( ∂ 21 ( ӽ ) , m 2 ( ∂ 21 ( ӽ ) ) − k 2 ( ∂ 32 ( ӽ ) ) ) .

Now

n 2 ( ∂ 32 ( ӽ ) ) – X ( ∂ 21 ( ӽ ) ) = 1 2 ∂ 21 ° ∂ 32 + ∂ 31 ° ∂ 32 − ∂ 32 ( 2 ) ° ∂ 21 ( ӽ ) = ( ∂ 21 ( ӽ ) ° ∂ 32 ( 2 ) + ∂ 31 ° ∂ 32 − ∂ 32 ( 2 ) ° ∂ 21 ) ( ӽ ) = ( ∂ 21 ( ӽ ) ° ∂ 32 ( 2 ) + ∂ 32 ° ∂ 31 − ∂ 21 ° ∂ 32 ( 2 ) − ∂ 32 ° ∂ 31 ) ( ӽ ) = 0 ,

m 1 ( ∂ 21 ( ӽ ) ) − k 2 ( ∂ 32 ( ӽ ) ) = 1 2 ∂ 32 ° ∂ 21 − ∂ 31 ° ∂ 21 ( ӽ ) − ∂ 21 ( 2 ) ° ∂ 32 ( ӽ ) = ( ∂ 32 ° ∂ 21 ( 2 ) − ∂ 21 ° ∂ 31 − ∂ 32 ° ∂ 21 ( 2 ) + ∂ 21 ° ∂ 31 ) ( ӽ ) = 0 .

Finally,

( σ 1 ° σ 2 ) ( ӽ 1 , ӽ 2 ) = σ 1 1 2 ∂ 21 ° ∂ 32 + ∂ 31 ( ӽ 1 ) − ∂ 32 ( 2 ) ( ӽ 2 ) , 1 2 ∂ 32 ° ∂ 21 − ∂ 31 ( ӽ 2 ) − ∂ 21 ( 2 ) ( ӽ 1 ) = ∂ 21 1 2 ∂ 21 ° ∂ 32 + ∂ 31 ( ӽ 1 ) − ∂ 21 ° ∂ 32 ( 2 ) ( ӽ 2 ) + ∂ 32 1 2 ∂ 32 ° ∂ 21 − ∂ 31 ( ӽ 2 ) − ∂ 32 ° ∂ 21 ( 2 ) ( ӽ 1 ) = ( ∂ 21 ( 2 ) ° ∂ 32 + ∂ 21 ° ∂ 31 − ∂ 32 ° ∂ 21 ( 2 ) ) ( ӽ 2 ) .

Then,

( σ 1 ° σ 2 ) ( ӽ 1 , ӽ 2 ) = ( ∂ 32 ° ∂ 21 ( 2 ) − ∂ 21 ° ∂ 31 + ∂ 21 ° ∂ 31 − ∂ 32 ° ∂ 21 ( 2 ) ( ӽ 1 ) + ( ∂ 21 ° ∂ 32 ( 2 ) + ∂ 32 ° ∂ 31 − ∂ 32 ° ∂ 31 − ∂ 21 ° ∂ 32 ( 2 ) ) ( ӽ 2 ) = 0 .

4 Conclusion

By using the diagrams, divided power, and Capelli’s identities we obtained the sequence of the terms of the characteristic zero is complex after we defined some maps to make these diagrams commute.

  1. Conflict of interest: Authors state no conflict of interest.

  2. Data availability statement: The most datasets generated and/or analysed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

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Received: 2023-06-10
Revised: 2023-08-02
Accepted: 2023-08-08
Published Online: 2024-01-18

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  146. Performance evaluation of grouted porous asphalt concrete
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https://doi.org/10.1515/eng-2022-0516
Keywords for this article
divided power; complex of Lascoux; Capelli’s identity
Creative Commons
BY 4.0

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  184. Evaluation of microstructure and mechanical properties of Al1050/Al2O3/Gr composite processed by forming operation ECAP
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  194. Evaluation of the effluent concentrations of Karbala wastewater treatment plant using reliability analysis
  195. Adsorbent made with inexpensive, local resources
  196. Effect of drain pipes on seepage and slope stability through a zoned earth dam
  197. Sediment accumulation in an 8 inch sewer pipe for a sample of various particles obtained from the streets of Karbala city, Iraq
  198. Special Issue: IETAS 2024 - Part I
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  202. The impact of using prestressed CFRP bars on the development of flexural strength
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  206. Special Issue: 51st KKBN - Part I
  207. Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection
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