Mixing
Why do different representations of “braid groups” give seemingly opposite results?
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One representation of "braid groups" is the Burau representation first propounded by Werner Burau in the 1930s. Later work has shown that this representation is "unfaithful" for n>=5, where n is the number of braids. That makes sense to me (a layman), insofar as the greater number of items, the greater the complexity, and the less "tractable" various formulations become.
But another representation of braid groups produced seemingly opposite results. In papers first published by Edward Formanek, and expanded by Inna Sysoeva, it was possible for Formanek to classify "irreducible complex representations B of Artin braid groups" to a dimension n-1. More to the point, Sysoeva showed that it was possible to classify these representations to a higher dimension, n, for n>=9. Here, the larger the n, the more manageable the braid groups are, and the more "tractable" the problem is.
I find these two results "paradoxical" to say the least. Is it, in fact the case, that while both results are true, one or both are counterintuitive? Or am I confusing seemingly similar concepts when in fact there is no relationship between them?
Or is it true that "nice properties" move in opposite directions in these two cases? To take a very simple example, 1< 1000 in the integer realm, but when you invert them to 1 and 1/1000, the first is greater than the second? (That is, you want to move toward more digits in the integer case, and toward fewer digits in the fractional case, if you want to maximize the number.)
braid-groups
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
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add a comment |
$begingroup$
One representation of "braid groups" is the Burau representation first propounded by Werner Burau in the 1930s. Later work has shown that this representation is "unfaithful" for n>=5, where n is the number of braids. That makes sense to me (a layman), insofar as the greater number of items, the greater the complexity, and the less "tractable" various formulations become.
But another representation of braid groups produced seemingly opposite results. In papers first published by Edward Formanek, and expanded by Inna Sysoeva, it was possible for Formanek to classify "irreducible complex representations B of Artin braid groups" to a dimension n-1. More to the point, Sysoeva showed that it was possible to classify these representations to a higher dimension, n, for n>=9. Here, the larger the n, the more manageable the braid groups are, and the more "tractable" the problem is.
I find these two results "paradoxical" to say the least. Is it, in fact the case, that while both results are true, one or both are counterintuitive? Or am I confusing seemingly similar concepts when in fact there is no relationship between them?
Or is it true that "nice properties" move in opposite directions in these two cases? To take a very simple example, 1< 1000 in the integer realm, but when you invert them to 1 and 1/1000, the first is greater than the second? (That is, you want to move toward more digits in the integer case, and toward fewer digits in the fractional case, if you want to maximize the number.)
braid-groups
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
$endgroup$
1
$begingroup$
I don't understand the question. What are the "opposite results" produced? Different representations of the same group can have very different properties.
$endgroup$
– Randall
Jan 13 at 2:07
$begingroup$
@Randall: The Burau representation has "nice properties" for n<=3, but it starts to "break down" (lose its "faithfulness" for n>5;4 is an indeterminnate case). on the other hand, Formanek believed that the irreducible complex representation for Artin braid groups existed only to dimension n-1; but his star student Sysoeva demonstrated that this was truly only for n<=8; if you made n 9 or higher, the property actually held all the way up to n dimensions. That is, the "nice properties" of the model improved when you raised n, the opposite of the preceding case.See the new last paragraph.
$endgroup$
– Tom Au
Jan 13 at 3:41
add a comment |
$begingroup$
One representation of "braid groups" is the Burau representation first propounded by Werner Burau in the 1930s. Later work has shown that this representation is "unfaithful" for n>=5, where n is the number of braids. That makes sense to me (a layman), insofar as the greater number of items, the greater the complexity, and the less "tractable" various formulations become.
But another representation of braid groups produced seemingly opposite results. In papers first published by Edward Formanek, and expanded by Inna Sysoeva, it was possible for Formanek to classify "irreducible complex representations B of Artin braid groups" to a dimension n-1. More to the point, Sysoeva showed that it was possible to classify these representations to a higher dimension, n, for n>=9. Here, the larger the n, the more manageable the braid groups are, and the more "tractable" the problem is.
I find these two results "paradoxical" to say the least. Is it, in fact the case, that while both results are true, one or both are counterintuitive? Or am I confusing seemingly similar concepts when in fact there is no relationship between them?
Or is it true that "nice properties" move in opposite directions in these two cases? To take a very simple example, 1< 1000 in the integer realm, but when you invert them to 1 and 1/1000, the first is greater than the second? (That is, you want to move toward more digits in the integer case, and toward fewer digits in the fractional case, if you want to maximize the number.)
braid-groups
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
$endgroup$
One representation of "braid groups" is the Burau representation first propounded by Werner Burau in the 1930s. Later work has shown that this representation is "unfaithful" for n>=5, where n is the number of braids. That makes sense to me (a layman), insofar as the greater number of items, the greater the complexity, and the less "tractable" various formulations become.
But another representation of braid groups produced seemingly opposite results. In papers first published by Edward Formanek, and expanded by Inna Sysoeva, it was possible for Formanek to classify "irreducible complex representations B of Artin braid groups" to a dimension n-1. More to the point, Sysoeva showed that it was possible to classify these representations to a higher dimension, n, for n>=9. Here, the larger the n, the more manageable the braid groups are, and the more "tractable" the problem is.
I find these two results "paradoxical" to say the least. Is it, in fact the case, that while both results are true, one or both are counterintuitive? Or am I confusing seemingly similar concepts when in fact there is no relationship between them?
Or is it true that "nice properties" move in opposite directions in these two cases? To take a very simple example, 1< 1000 in the integer realm, but when you invert them to 1 and 1/1000, the first is greater than the second? (That is, you want to move toward more digits in the integer case, and toward fewer digits in the fractional case, if you want to maximize the number.)
braid-groups
braid-groups
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
edited Jan 13 at 3:40
Tom Au
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
asked Jan 12 at 19:51
Tom AuTom Au
1,2701333
1,2701333
1
$begingroup$
I don't understand the question. What are the "opposite results" produced? Different representations of the same group can have very different properties.
$endgroup$
– Randall
Jan 13 at 2:07
$begingroup$
@Randall: The Burau representation has "nice properties" for n<=3, but it starts to "break down" (lose its "faithfulness" for n>5;4 is an indeterminnate case). on the other hand, Formanek believed that the irreducible complex representation for Artin braid groups existed only to dimension n-1; but his star student Sysoeva demonstrated that this was truly only for n<=8; if you made n 9 or higher, the property actually held all the way up to n dimensions. That is, the "nice properties" of the model improved when you raised n, the opposite of the preceding case.See the new last paragraph.
$endgroup$
– Tom Au
Jan 13 at 3:41
add a comment |
1
$begingroup$
I don't understand the question. What are the "opposite results" produced? Different representations of the same group can have very different properties.
$endgroup$
– Randall
Jan 13 at 2:07
$begingroup$
@Randall: The Burau representation has "nice properties" for n<=3, but it starts to "break down" (lose its "faithfulness" for n>5;4 is an indeterminnate case). on the other hand, Formanek believed that the irreducible complex representation for Artin braid groups existed only to dimension n-1; but his star student Sysoeva demonstrated that this was truly only for n<=8; if you made n 9 or higher, the property actually held all the way up to n dimensions. That is, the "nice properties" of the model improved when you raised n, the opposite of the preceding case.See the new last paragraph.
$endgroup$
– Tom Au
Jan 13 at 3:41
1
1
$begingroup$
I don't understand the question. What are the "opposite results" produced? Different representations of the same group can have very different properties.
$endgroup$
– Randall
Jan 13 at 2:07
$begingroup$
I don't understand the question. What are the "opposite results" produced? Different representations of the same group can have very different properties.
$endgroup$
– Randall
Jan 13 at 2:07
$begingroup$
@Randall: The Burau representation has "nice properties" for n<=3, but it starts to "break down" (lose its "faithfulness" for n>5;4 is an indeterminnate case). on the other hand, Formanek believed that the irreducible complex representation for Artin braid groups existed only to dimension n-1; but his star student Sysoeva demonstrated that this was truly only for n<=8; if you made n 9 or higher, the property actually held all the way up to n dimensions. That is, the "nice properties" of the model improved when you raised n, the opposite of the preceding case.See the new last paragraph.
$endgroup$
– Tom Au
Jan 13 at 3:41
$begingroup$
@Randall: The Burau representation has "nice properties" for n<=3, but it starts to "break down" (lose its "faithfulness" for n>5;4 is an indeterminnate case). on the other hand, Formanek believed that the irreducible complex representation for Artin braid groups existed only to dimension n-1; but his star student Sysoeva demonstrated that this was truly only for n<=8; if you made n 9 or higher, the property actually held all the way up to n dimensions. That is, the "nice properties" of the model improved when you raised n, the opposite of the preceding case.See the new last paragraph.
$endgroup$
– Tom Au
Jan 13 at 3:41
add a comment |
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$begingroup$
I don't understand the question. What are the "opposite results" produced? Different representations of the same group can have very different properties.
$endgroup$
– Randall
Jan 13 at 2:07
$begingroup$
@Randall: The Burau representation has "nice properties" for n<=3, but it starts to "break down" (lose its "faithfulness" for n>5;4 is an indeterminnate case). on the other hand, Formanek believed that the irreducible complex representation for Artin braid groups existed only to dimension n-1; but his star student Sysoeva demonstrated that this was truly only for n<=8; if you made n 9 or higher, the property actually held all the way up to n dimensions. That is, the "nice properties" of the model improved when you raised n, the opposite of the preceding case.See the new last paragraph.
$endgroup$
– Tom Au
Jan 13 at 3:41