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I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:

$$e^{t*\chi_{Adj}(a)}$$

where $a$ is for example $SU(N)$ matrix. I would like to expand it in series of $t$ and compute coefficient in from of singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition. I take an adjoint representation of SU(2)

G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()

and try to feed it into symbolic expression

exp(t*ad)

The result is TypeError. There are two problems both due to the TypeError that I see

1) While ad**n gives me write decomposition if I try the following

t = var('t')
g(t) = t**2
g(ad)

I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.

2) The same happens if I just try to multiply ad with some symbolic variables

t*ad

The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters inside expression containing other symbolic variables and powers of representations and expand in irreps.

I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:

$$e^{t*\chi_{Adj}(a)}$$

where $a$ is for example $SU(N)$ matrix. I would like to expand it in series of $t$ and compute coefficient in from of of the singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition. I take an adjoint representation of SU(2)

G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()

and try to feed it into symbolic expression

exp(t*ad)

The result is TypeError. There are two problems both due to the TypeError that I see

1) While ad**n gives me write decomposition if I try the following

t = var('t')
g(t) = t**2
g(ad)

I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.

2) The same happens if I just try to multiply ad with some symbolic variables

t*ad

The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters inside expression containing other symbolic variables and powers of representations and expand in irreps. irreps.

I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:

$$e^{t*\chi_{Adj}(a)}$$

where $a$ is for example $SU(N)$ matrix. I would like to expand it in series of $t$ and compute coefficient of the singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition. decomposition in irreps. I take an adjoint representation of SU(2)

G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()

and try to feed it into symbolic expression

exp(t*ad)

The result is TypeError. There are two problems both due to the TypeError that I see

1) While ad**n gives me write decomposition if I try the following

t = var('t')
g(t) = t**2
g(ad)

I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.

2) The same happens if I just try to multiply ad with some symbolic variables

t*ad

The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters inside expression containing other symbolic variables and powers of representations and expand in irreps.

I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:

$$e^{t*\chi_{Adj}(a)}$$

where $a$ is for example $SU(N)$ matrix. I would like to expand it in series of $t$ and compute coefficient of the singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition in irreps. irreps and it seems a natural choice for this problem. To solve my problem I take an adjoint representation of SU(2)

G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()

and try to feed it into symbolic expression

exp(t*ad)

The result is TypeError. There are two problems both due to the TypeError that I see

1) While ad**n gives me write decomposition if I try the following

t = var('t')
g(t) = t**2
g(ad)

I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.

2) The same happens if I just try to multiply ad with some symbolic variables

t*ad

The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters inside expression containing other symbolic variables and powers of representations and expand in irreps.

I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:

$$e^{t*\chi_{Adj}(a)}$$

where $a$ is for example $SU(N)$ $SU(2)$ matrix. I would like to expand it in series of $t$ and compute coefficient of the singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition in irreps and it seems a natural choice for this problem. To solve my problem I take an adjoint representation of SU(2)

G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()

and try to feed it into symbolic expression

exp(t*ad)

The result is TypeError. There are two problems both due to the TypeError that I see

1) While ad**n gives me write decomposition if I try the following

t = var('t')
g(t) = t**2
g(ad)

I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.

2) The same happens if I just try to multiply ad with some symbolic variables

t*ad

The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters inside expression containing other symbolic variables and powers of representations and expand in irreps.

I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:

$$e^{t*\chi_{Adj}(a)}$$

where $a$ is for example $SU(2)$ matrix. I would like to expand it in series of $t$ and compute coefficient of the singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition in irreps and it seems a natural choice for this problem. To solve my problem I take an adjoint representation of SU(2)

G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()

and try to feed it into symbolic expression

exp(t*ad)

The result is TypeError. There are two problems both due to the TypeError that I see

1) While ad**n gives me write decomposition if I try the following

t = var('t')
g(t) = t**2
g(ad)

I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.

2) The same happens if I just try to multiply ad with some symbolic variables

t*ad

The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters inside into an expression containing other symbolic variables and powers of representations and expand in irreps.