Fdtxjr

How to Reset Passwords on Multiple Websites Easily?

Implement the Thanos sorting algorithm

Pole-zeros of a real-valued causal FIR system

CREATE opcode: what does it really do?

India just shot down a satellite from the ground. At what altitude range is the resulting debris field?

Increase performance creating Mandelbrot set in python

What is paid subscription needed for in Mortal Kombat 11?

Failed to fetch jessie backports repository

Is this apparent Class Action settlement a spam message?

Inappropriate reference requests from Journal reviewers

Do the temporary hit points from the Battlerager barbarian's Reckless Abandon stack if I make multiple attacks on my turn?

Is there a good way to store credentials outside of a password manager?

Term for the "extreme-extension" version of a straw man fallacy?

Why Were Madagascar and New Zealand Discovered So Late?

Is exact Kanji stroke length important?

How can I kill an app using Terminal?

Go Pregnant or Go Home

How does Loki do this?

Customer Requests (Sometimes) Drive Me Bonkers!

Is there a problem with hiding "forgot password" until it's needed?

Avoiding estate tax by giving multiple gifts

A particular customize with green line and letters for subfloat

Trouble understanding the speech of overseas colleagues

Gears on left are inverse to gears on right?

Simple formula for the dimension of weight spaces of Verma module?

irreducible highest weight modulesProperties of the longest element in a Weyl groupHow to use Weyl dimension formula?Weight spaces of Verma modulesInducing highest weight modulesDimension of a weight space which is of weight $0$.How to find the multiplicity of weight in a Verma module?Composition series for Verma modules.finding high weight vector in Verma moduleThe relation between Weyl character formula and Frobenius characteristic map

0

$begingroup$

Let $mathfrakg$ be a simple Lie algebra (e.g. $mathfraksl_n$), and let $M_lambda$ be the Verma module with highest weight $lambda$.
Is there a simple formula for $dim (M_lambda)_mu$, where $(M_lambda)_mu$ denotes the weight space with weight $mu$? Of course Weyl character formula gives me the generating function, but I am looking for a closed form formula for $dim (M_lambda)_mu$.

---

Edit : Apparently this is called the Kostant partition function.

representation-theory lie-algebras verma-modules

share|cite|improve this question

edited Mar 21 at 6:21

Saad

20.2k92352

asked Mar 17 at 22:26

HenryHenry

1,731726

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which formulation of Weyl's character formula do you mean? I recall it usually being given directly, rather than via a generating function.
  $endgroup$
  – Tobias Kildetoft
  Mar 18 at 17:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @TobiasKildetoft For instance, if you expand $1/Delta(z)$ into a power series where $Delta(z)$ is a Weyl denominator, the coefficients of this series are the certain dimensions that I want.
  $endgroup$
  – Henry
  Mar 19 at 19:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As you write in your edit, it's the Kostant partition function: the number of ways to write $lambda-mu$ as a non-negative integer linear combination of positive roots. You won't get a better formula than that in general.
  $endgroup$
  – Stephen
  Mar 20 at 22:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Stephen But isn't Kostant partition function piecewise polynomial? If an explicit form of the piecewise polynomial is known for, say, $A_n$, then that would be helpful.
  $endgroup$
  – Henry
  Mar 20 at 23:18
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $mathfrakg$ be a simple Lie algebra (e.g. $mathfraksl_n$), and let $M_lambda$ be the Verma module with highest weight $lambda$.
Is there a simple formula for $dim (M_lambda)_mu$, where $(M_lambda)_mu$ denotes the weight space with weight $mu$? Of course Weyl character formula gives me the generating function, but I am looking for a closed form formula for $dim (M_lambda)_mu$.

---

Edit : Apparently this is called the Kostant partition function.

representation-theory lie-algebras verma-modules

share|cite|improve this question

edited Mar 21 at 6:21

Saad

20.2k92352

asked Mar 17 at 22:26

HenryHenry

1,731726

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which formulation of Weyl's character formula do you mean? I recall it usually being given directly, rather than via a generating function.
  $endgroup$
  – Tobias Kildetoft
  Mar 18 at 17:01
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @TobiasKildetoft For instance, if you expand $1/Delta(z)$ into a power series where $Delta(z)$ is a Weyl denominator, the coefficients of this series are the certain dimensions that I want.
  $endgroup$
  – Henry
  Mar 19 at 19:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  As you write in your edit, it's the Kostant partition function: the number of ways to write $lambda-mu$ as a non-negative integer linear combination of positive roots. You won't get a better formula than that in general.
  $endgroup$
  – Stephen
  Mar 20 at 22:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Stephen But isn't Kostant partition function piecewise polynomial? If an explicit form of the piecewise polynomial is known for, say, $A_n$, then that would be helpful.
  $endgroup$
  – Henry
  Mar 20 at 23:18
  

  
  
  
  

add a comment | 

$begingroup$

Let $mathfrakg$ be a simple Lie algebra (e.g. $mathfraksl_n$), and let $M_lambda$ be the Verma module with highest weight $lambda$.
Is there a simple formula for $dim (M_lambda)_mu$, where $(M_lambda)_mu$ denotes the weight space with weight $mu$? Of course Weyl character formula gives me the generating function, but I am looking for a closed form formula for $dim (M_lambda)_mu$.

---

Edit : Apparently this is called the Kostant partition function.

representation-theory lie-algebras verma-modules

share|cite|improve this question

edited Mar 21 at 6:21

Saad

20.2k92352

asked Mar 17 at 22:26

HenryHenry

1,731726

$endgroup$

share|cite|improve this question

edited Mar 21 at 6:21

Saad

20.2k92352

asked Mar 17 at 22:26

HenryHenry

1,731726

share|cite|improve this question

edited Mar 21 at 6:21

Saad

20.2k92352

asked Mar 17 at 22:26

HenryHenry

1,731726

edited Mar 21 at 6:21

Saad

20.2k92352

edited Mar 21 at 6:21

Saad

20.2k92352

asked Mar 17 at 22:26

HenryHenry

1,731726

asked Mar 17 at 22:26

HenryHenry

1,731726