Making sense of classical Lie groups
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Andrius Kulikauskas
May 18, 2016, 12:50:25 PM5/18/16
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Kirby,
Happy birthday!
Thank you for the many images. It was fun to see one in the Mitchell
lecture from the self-declared Republic of Uzupis, across the river,
here in Vilnius, Lithuania. And also to read your post about Vilnius.
And for all the links and ideas.
Today I watched video "J2 Unitary Groups" from doctorphys series on
"Theoretical Physics" at YouTube.
https://www.youtube.com/playlist?list=PL54DF0652B30D99A4
I appreciate that he actually calculates some small, concrete examples,
which is what I try to do, too. And then he made an extra remark which
made things click for me. He explained for a particular matrix that
when you take the inverse, it's just like reversing the direction of the
angle.
So I want to apply that insight and write up my thoughts on making sense
of the classical Lie groups. I will start by describing what they are.
First, I will explain what a "group" is. In math, a group is at
work/play whenever we think/talk about actions. For example, imagine a
drawing in the plane. Let's establish some point in the drawing where
we pin it down to the plane. Then we can rotate that picture by X
degrees. Let's say, for the sake of concreteness, that X is an integer
from 0 to 359. Then the rotations are "actions" in that they can be:
* added: rotations by X and by Y can be added to get a rotation by X+Y
* it's associative (you can insert parentheses as you prefer): rotating
by ((X+Y)+Z) = rotating by (X + (Y+Z))
* there's an action which does nothing, leaves things be, namely the
"identity" action 0
* you can undo each action. Rotation by X (say 40 degrees) can be undone
by some other rotation (320 degrees = -40 degrees).
We call this a "group" of actions (elements, operators, etc.)
You can have subgroups. So if we restrict ourselves to rotations by
multiples of 5 degrees (5, 10, 15...) we will have the 60 rotations we
need for the minute hand of an old-fashioned, pre-digital clock. If we
restrict ourselves to rotations of 30 degrees (30, 60, 90...) we will
have the 12 rotations we need for the hour hand of that same clock. If
we restrict to rotations of 90 degrees (0, 90, 180, 270), then we have
the 4 rotations which would keep our drawing unchanged if it was a
square. This is called a "symmetry group" but the others are symmetry
groups, too, for the right objects. If we restrict to rotations of 72
degrees, then we have the 5 rotations that would keep a pentagon
unchanged. This last subgroup would be special because it doesn't have
any subgroups, partly because 5 is a prime number. Such subgroups are
valued as building blocks for more complicated groups.
All of these groups are "commutative" because rotating by X and by Y is
the same as rotating by Y and then by X. But there are groups which are
not commutative. Let's take the 4 rotations (0, 90, 180, 270) of the
square in the plane and let's add a reflection R that flips the square
over on that plane. Then it turns out that we have a group of 8 actions
and we have, for example, that 90 + R does not equal R + 90. You can
imagine non-commutativity is typical when you play with a Rubik's cube
(the order of the actions matters).
All of these groups are finite. But we can also have infinite groups.
Imagine if we rotated by any real number of degrees. These rotations
happen to also be continuous, which is not trivial to make rigorous, but
I think basically here boils down to the fact that we can make
infinitesimal, that is, itsy bitsy rotations, as small as we want. So
then we have a Lie group. You can imagine that Lie groups are important
in physics because we live in a world where actions can be subtle and
continuous.
The "building blocks" of the Lie groups have been classified. There are
four families of groups, An, Bn, Cn, Dn, where n is any natural number.
These are called the classical Lie groups. There are also five
exceptional Lie groups. I would like to intuitively, qualitatively
understand the essence of those classical groups, so I could feel what
makes them different and what they share in common. I have failed to
find any such exposition.
But I think I'm getting closer. These classical Lie groups are known as
the unitary (An), orthogonal (odd and even, Bn and Dn) and symplectic
(Cn). These can all be thought of as groups of matrices. Each matrix
can be thought of as an action, as a rule which tells you how to break
up one vector into components, modify those components, and then output
a new vector. These rules can be composed just like actions. In
general, matrices and matrix multiplication are used to describe
explicitly a group's actions and how they are composed. This is called
a group representation. It's a bit like writing down the multiplication
table of a group. Mathematicians study the restrictions on the types of
tables possible and can determine from that the nature of the group, for
example, how it breaks down into subgroups.
The numbers in these matrices can be complex numbers. Now for me the
key distinction seems to be that in each group there is a special way to
imagine how an action is undone. In other words, there is a special
relation between a matrix and its inverse.
You undo an action:
* in a unitary group, by taking the conjugate transpose of its matrix.
* in an orthogonal group, by taking the symmetric transpose of its matrix.
* in a symplectic group, by taking the anti-symmetric transpose of its
matrix.
What is a transpose? An NxN matrix is a set of rules which takes a
column vector (of N components) as the input and outputs a row vector
(of N components). Then the transpose is the same set of rules but just
reorganized so that the input is a row vector and the matrix outputs a
column vector. That will be important if we think in terms of tensors
where the row vectors and the column vectors are the extremes of
top-down thinking and bottom-up thinking in building a coordinate
space. So this is what the classical Lie groups all have in common.
Where they differ is on how they modify the transpose so that the
group's action is undone.
I will think about that and write in the days to come.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Happy birthday!
Thank you for the many images. It was fun to see one in the Mitchell
lecture from the self-declared Republic of Uzupis, across the river,
here in Vilnius, Lithuania. And also to read your post about Vilnius.
And for all the links and ideas.
Today I watched video "J2 Unitary Groups" from doctorphys series on
"Theoretical Physics" at YouTube.
https://www.youtube.com/playlist?list=PL54DF0652B30D99A4
I appreciate that he actually calculates some small, concrete examples,
which is what I try to do, too. And then he made an extra remark which
made things click for me. He explained for a particular matrix that
when you take the inverse, it's just like reversing the direction of the
angle.
So I want to apply that insight and write up my thoughts on making sense
of the classical Lie groups. I will start by describing what they are.
First, I will explain what a "group" is. In math, a group is at
work/play whenever we think/talk about actions. For example, imagine a
drawing in the plane. Let's establish some point in the drawing where
we pin it down to the plane. Then we can rotate that picture by X
degrees. Let's say, for the sake of concreteness, that X is an integer
from 0 to 359. Then the rotations are "actions" in that they can be:
* added: rotations by X and by Y can be added to get a rotation by X+Y
* it's associative (you can insert parentheses as you prefer): rotating
by ((X+Y)+Z) = rotating by (X + (Y+Z))
* there's an action which does nothing, leaves things be, namely the
"identity" action 0
* you can undo each action. Rotation by X (say 40 degrees) can be undone
by some other rotation (320 degrees = -40 degrees).
We call this a "group" of actions (elements, operators, etc.)
You can have subgroups. So if we restrict ourselves to rotations by
multiples of 5 degrees (5, 10, 15...) we will have the 60 rotations we
need for the minute hand of an old-fashioned, pre-digital clock. If we
restrict ourselves to rotations of 30 degrees (30, 60, 90...) we will
have the 12 rotations we need for the hour hand of that same clock. If
we restrict to rotations of 90 degrees (0, 90, 180, 270), then we have
the 4 rotations which would keep our drawing unchanged if it was a
square. This is called a "symmetry group" but the others are symmetry
groups, too, for the right objects. If we restrict to rotations of 72
degrees, then we have the 5 rotations that would keep a pentagon
unchanged. This last subgroup would be special because it doesn't have
any subgroups, partly because 5 is a prime number. Such subgroups are
valued as building blocks for more complicated groups.
All of these groups are "commutative" because rotating by X and by Y is
the same as rotating by Y and then by X. But there are groups which are
not commutative. Let's take the 4 rotations (0, 90, 180, 270) of the
square in the plane and let's add a reflection R that flips the square
over on that plane. Then it turns out that we have a group of 8 actions
and we have, for example, that 90 + R does not equal R + 90. You can
imagine non-commutativity is typical when you play with a Rubik's cube
(the order of the actions matters).
All of these groups are finite. But we can also have infinite groups.
Imagine if we rotated by any real number of degrees. These rotations
happen to also be continuous, which is not trivial to make rigorous, but
I think basically here boils down to the fact that we can make
infinitesimal, that is, itsy bitsy rotations, as small as we want. So
then we have a Lie group. You can imagine that Lie groups are important
in physics because we live in a world where actions can be subtle and
continuous.
The "building blocks" of the Lie groups have been classified. There are
four families of groups, An, Bn, Cn, Dn, where n is any natural number.
These are called the classical Lie groups. There are also five
exceptional Lie groups. I would like to intuitively, qualitatively
understand the essence of those classical groups, so I could feel what
makes them different and what they share in common. I have failed to
find any such exposition.
But I think I'm getting closer. These classical Lie groups are known as
the unitary (An), orthogonal (odd and even, Bn and Dn) and symplectic
(Cn). These can all be thought of as groups of matrices. Each matrix
can be thought of as an action, as a rule which tells you how to break
up one vector into components, modify those components, and then output
a new vector. These rules can be composed just like actions. In
general, matrices and matrix multiplication are used to describe
explicitly a group's actions and how they are composed. This is called
a group representation. It's a bit like writing down the multiplication
table of a group. Mathematicians study the restrictions on the types of
tables possible and can determine from that the nature of the group, for
example, how it breaks down into subgroups.
The numbers in these matrices can be complex numbers. Now for me the
key distinction seems to be that in each group there is a special way to
imagine how an action is undone. In other words, there is a special
relation between a matrix and its inverse.
You undo an action:
* in a unitary group, by taking the conjugate transpose of its matrix.
* in an orthogonal group, by taking the symmetric transpose of its matrix.
* in a symplectic group, by taking the anti-symmetric transpose of its
matrix.
What is a transpose? An NxN matrix is a set of rules which takes a
column vector (of N components) as the input and outputs a row vector
(of N components). Then the transpose is the same set of rules but just
reorganized so that the input is a row vector and the matrix outputs a
column vector. That will be important if we think in terms of tensors
where the row vectors and the column vectors are the extremes of
top-down thinking and bottom-up thinking in building a coordinate
space. So this is what the classical Lie groups all have in common.
Where they differ is on how they modify the transpose so that the
group's action is undone.
I will think about that and write in the days to come.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Joseph Austin
May 18, 2016, 10:16:55 PM5/18/16
to mathf...@googlegroups.com
On May 18, 2016, at 12:21 PM, Andrius Kulikauskas <m...@ms.lt> wrote:So I want to apply that insight and write up my thoughts on making sense of the classical Lie groups.
Andrius,
I appreciate your attempting to make this important subject understandable.
I'm looking forward to your further explications.
Joe Austin
kirby urner
May 19, 2016, 10:45:21 AM5/19/16
to mathf...@googlegroups.com
On Wed, May 18, 2016 at 9:21 AM, Andrius Kulikauskas <m...@ms.lt> wrote:
Kirby,
Happy birthday!
Thank you. I got to do many fun things on my birthday, and I'm still celebrating!
Thank you for the many images. It was fun to see one in the Mitchell lecture from the self-declared Republic of Uzupis, across the river, here in Vilnius, Lithuania. And also to read your post about Vilnius. And for all the links and ideas.
That's some kind of art colony, that Uz? I enjoy art colonies that play with the nation-state meme.
Here in Oregon we have Rogue Nation, which styles itself as a nation whereas of course no UN map would recognize such a place. They make beer. I sometimes refer to Python Nation in the same spirit, given we have a benevolent dictator (Guido). I've referred to myself as Minister of Education (with portfolio) in that ecosystem.
Here in Oregon we have Rogue Nation, which styles itself as a nation whereas of course no UN map would recognize such a place. They make beer. I sometimes refer to Python Nation in the same spirit, given we have a benevolent dictator (Guido). I've referred to myself as Minister of Education (with portfolio) in that ecosystem.
Thank you for your presentation regarding groups and Lie groups in particular. I started watching doctorphys videos and got to his earliest mention of Group Theory way back at A7 and A8, where he introduces the "military group" with operations for group elements, and the binary "then" for the group operation e.g. "Attention! then Left!"; "Attention! the 'Boutface!"
{A!, L!, R!, B!} are the four operations (set members), the group operation "then". If I go Left! then Right! I'm back where I started. That's the same as just saying A! in this picture, but then are we allowed to perform unary operations? I have the same question about inverse (reciprocal) or negation in groups. They always dwell on just the set members and a binary op, but the unary op of negation always seems to creep in, sort of under the rug.
{A!, L!, R!, B!} are the four operations (set members), the group operation "then". If I go Left! then Right! I'm back where I started. That's the same as just saying A! in this picture, but then are we allowed to perform unary operations? I have the same question about inverse (reciprocal) or negation in groups. They always dwell on just the set members and a binary op, but the unary op of negation always seems to creep in, sort of under the rug.
I found it humorous that Attention! would be considered a "no op" i.e. the "makes no difference" action. What does that say about soldiers eh?
A soldier may turn 90 degrees left (L!) or 90 degrees right (R!) or turn about B! (like in your rotation-based example), but the command "A!" brings on no change in state, not in a way we care about. A! A! A! A! A! = A! = L! R! (A! is a member, not a unary operation -- a bit confusing as we imagine these member operations as actions on a soldier).
A soldier may turn 90 degrees left (L!) or 90 degrees right (R!) or turn about B! (like in your rotation-based example), but the command "A!" brings on no change in state, not in a way we care about. A! A! A! A! A! = A! = L! R! (A! is a member, not a unary operation -- a bit confusing as we imagine these member operations as actions on a soldier).
That got me to thinking of "soldier as state machine" and how "Attention!" could be like an "on switch" such that L!, R! and B! are only possible when the soldier is "at attention" (on).
The inverse or undo operation would be "At Ease!" (off) -- we could use Ease! for a unique letter E!. {A!, E!, L!, R!, B!} would now be the new set.
The inverse or undo operation would be "At Ease!" (off) -- we could use Ease! for a unique letter E!. {A!, E!, L!, R!, B!} would now be the new set.
However, now it seems we have no identity member. A! then E! = E! i.e. if we say "Attention!" then "At Ease!", the solider will be "at ease" (off). E! then A! leaves the soldier off. So (A! then E!) is not equal to (E! then A!) which means neither element plays the role of identity.
(A! then E! then L!) = (A! then E!) because L! has no effect when the soldier is turned off. (E! then A! then L!) on the other hand, will result in a left turning, because as A! right before L! makes L! effective.
(A! then E! then L!) = (A! then E!) because L! has no effect when the soldier is turned off. (E! then A! then L!) on the other hand, will result in a left turning, because as A! right before L! makes L! effective.
Is this any longer an associative set?
((E! then A!) then L!) results in a left turn (90 degrees).
(E! then (A! then L!)) makes the left turn happen, however the word "then" is problematic as it rubs against our wanting to apply E! "after" or "later". ((A! then L!) then E!) is, however, not a test of associativity vis-a-vis ((E! then A!) then L!).
The issue is parenthesis tell us "what to do first" and yet an operation named "then" also implies order and these two may therefore send conflicting messages. B! after L! == L! then B!. It's like we almost have two operations {then, after}, but then could say one is syntactic sugar for the other.
((E! then A!) then L!) results in a left turn (90 degrees).
(E! then (A! then L!)) makes the left turn happen, however the word "then" is problematic as it rubs against our wanting to apply E! "after" or "later". ((A! then L!) then E!) is, however, not a test of associativity vis-a-vis ((E! then A!) then L!).
The issue is parenthesis tell us "what to do first" and yet an operation named "then" also implies order and these two may therefore send conflicting messages. B! after L! == L! then B!. It's like we almost have two operations {then, after}, but then could say one is syntactic sugar for the other.
Even with the original military group, ((R! then R!) then B!) == (R! then (R! then B!)) seems problematic. What exactly does that mean i.e. how do we do (R! then B!) first and still obey "then"?
To make {A!, E!, L!, R!, B!} a group again, I supposed we'd need to add N! for "do nothing!". {A!, E!, L!, R!, B!, N!} is now a group under the operation "then" although allowing "then" as an operator introduces time, which fights against or concept of testing associativity; an issue with the military group too it seems. Just having a Cayley table would answer all questions.
Kirby
I will think about that and write in the days to come.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
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kirby urner
May 19, 2016, 11:10:24 AM5/19/16
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On Thu, May 19, 2016 at 7:45 AM, kirby urner <kirby...@gmail.com> wrote:
However, now it seems we have no identity member. A! then E! = E! i.e. if we say "Attention!" then "At Ease!", the solider will be "at ease" (off). E! then A! leaves the soldier off.
My bad: E! then A! of course leave the soldier in an attentive state (on, and ready to obey additional commands).
Back to quadrays, I do a similar investigation of the idea of basis vectors in a vector space. I consider negation, the act of flipping a vector 180 degrees, to be a unary operation. -k = ---k (one 180 flip equals three flips). k = --k (two 180 flips equals no flips -- like B! then B! in the military group).
In the shoptalk of linear algebra, we say i, j, k all pointing in the positive direction are members of the basis set in XYZ, but -i, -j, -k are not. The negative mirrors are second class citizens in not having "basis" status. Only the positives get that singular honor.
Scalar multiplication by -1 creates a non-basis vector that is nevertheless critical in reaching a point. If we had only positive scalars, we could still shrink a basis vector to 1/8th its size, or expand it 8 times, but we could not flip it. Any point in (+ + +) could be reached by tip-to-tail vector addition.
In the shoptalk of linear algebra, we say i, j, k all pointing in the positive direction are members of the basis set in XYZ, but -i, -j, -k are not. The negative mirrors are second class citizens in not having "basis" status. Only the positives get that singular honor.
Scalar multiplication by -1 creates a non-basis vector that is nevertheless critical in reaching a point. If we had only positive scalars, we could still shrink a basis vector to 1/8th its size, or expand it 8 times, but we could not flip it. Any point in (+ + +) could be reached by tip-to-tail vector addition.
We have two ways of introducing XYZ don't we?
The "fly buzzing in the corner of the room" encourages us to think of the fly as a roving point P, with perpendiculars going to the three planes of that corner: room ceiling, room wall, room wall. These are the XY, YZ and XZ planes, where the ceiling is XY and Z is the vertical edge where the two walls meet.
The "fly buzzing in the corner of the room" encourages us to think of the fly as a roving point P, with perpendiculars going to the three planes of that corner: room ceiling, room wall, room wall. These are the XY, YZ and XZ planes, where the ceiling is XY and Z is the vertical edge where the two walls meet.
How the fly projected on three "screens" (planes) gives the fly's unique coordinates may seem a bit mysterious at first, but then we realize that there's only so much Xness, Yness and Zness in every point. A fly on the ceiling has no Zness at all. That's our second way of getting to any point P: we move only in basis vector directions for a specific distance, adding these "scaled" vectors tip-to-tail. By this means, we map to any P as a vector sum.
Quadrays are just like this, just the three edges are splayed apart a bit more i.e. the angles are not 90-90-90. Lets call these X', Y' and Z' instead. The change of angles makes no difference to the language game at hand. There's still a unique vector sum of X' Y' Z' in this all positive octant (except now it's a quadrant).
A quadrant because just one more vector, W', will finish dividing space into four equivalent volumes so the fly is able to treat the corner as an origin, the walls and ceiling no long solid. Only the edges remain. The fly may fly completely around the corner, now the origin.
In XYZ, we don't just add W, but rather three more edges in directions opposite to X, Y and Z. We use a "jack" instead of a "caltrop". We add these additional directions by means of scalar negation, flipping 180. With quadrays, W' is considered another basis vector and all of these vectors only need positive scalar operations to grow or shrink in the direction already pointed. Shrink to 0, extend as far as you like. X' + Y' + Z' + W' = (X', Y', Z', W') i.e. any point P, with tip-to-tail addition done the same way.
In XYZ, we don't just add W, but rather three more edges in directions opposite to X, Y and Z. We use a "jack" instead of a "caltrop". We add these additional directions by means of scalar negation, flipping 180. With quadrays, W' is considered another basis vector and all of these vectors only need positive scalar operations to grow or shrink in the direction already pointed. Shrink to 0, extend as far as you like. X' + Y' + Z' + W' = (X', Y', Z', W') i.e. any point P, with tip-to-tail addition done the same way.
Kirby
Joseph Austin
May 19, 2016, 9:09:16 PM5/19/16
to mathf...@googlegroups.com
On May 19, 2016, at 10:45 AM, kirby urner <kirby...@gmail.com> wrote:That got me to thinking of "soldier as state machine" and how "Attention!" could be like an "on switch" such that L!, R! and B! are only possible when the soldier is "at attention" (on).
Kirby,
I've been thinks some about the relationships between groups (defined axiomatically as a binary operation on sets) vs. state machines, and procedural vs. object-oriented programming languages. As you suggest, the "state machine" approach (or OO approach) reduces binary operations to unary by virtue of incorporating one of the operands of the binary operator into a retained state.
I've also observed that most "physical" operations are conservative, merely rearranging rather than creating or destroying "objects", so that suggests we explore a kind of "math" more concerned with arrangements or shapes than with quantities or counts. (The physical quantities and counts must always balance on both sides of an "interaction".) In this context, the permutation group and it's subgroups (and continuous analogues?) seems like a reasonable place to start.
What would "math" look like if the focus were "zero-sum" transactions and cycles of groups, instead of computing functions of variables and "solving" for inverses?
Joe Austin
kirby urner
May 20, 2016, 10:44:59 AM5/20/16
to mathf...@googlegroups.com
On Thu, May 19, 2016 at 6:09 PM, Joseph Austin <drtec...@gmail.com> wrote:
On May 19, 2016, at 10:45 AM, kirby urner <kirby...@gmail.com> wrote:That got me to thinking of "soldier as state machine" and how "Attention!" could be like an "on switch" such that L!, R! and B! are only possible when the soldier is "at attention" (on).Kirby,I've been thinks some about the relationships between groups (defined axiomatically as a binary operation on sets) vs. state machines, and procedural vs. object-oriented programming languages. As you suggest, the "state machine" approach (or OO approach) reduces binary operations to unary by virtue of incorporating one of the operands of the binary operator into a retained state.
Yes, the concepts seem rather subtle.
If we take A-Z plus space (27 characters in all) to a reordering of same, such that A -> S, B -> Q, C ->E..., with every letter paired with its "outcome" in an unambiguously reversible way, we have the makings of a group.
Such permutations may be composed (the group's binary op, similar to "then" in the military group) and each has an inverse that "undoes" the scramble another does. S -> A, Q -> B, E -> C and so on.
Such permutations may be composed (the group's binary op, similar to "then" in the military group) and each has an inverse that "undoes" the scramble another does. S -> A, Q -> B, E -> C and so on.
Consider all strings made of A-Z plus space such as "THE RAIN IN SPAIN". That's not a permutation, but may be "acted upon" by a permutation such that what comes out is a corresponding string with each letter (and space) replaced per said permutation.
There's a theorem that all finite groups may be modeled as permutations i.e. as elements being re-arranged. Your intuition that that's the right place to start seems dead on:
I've also observed that most "physical" operations are conservative, merely rearranging rather than creating or destroying "objects", so that suggests we explore a kind of "math" more concerned with arrangements or shapes than with quantities or counts. (The physical quantities and counts must always balance on both sides of an "interaction".) In this context, the permutation group and it's subgroups (and continuous analogues?) seems like a reasonable place to start.
Yes, perfect. Back to my three chords (as cued by P. Farrell), "data structures" is one of them and a permutation is an ideal example of the data structure we call a "mapping". {"A":"S", "B":"Q", "C":"E"...} would be typical syntax with "A":"S" signifying that any occurrence of "A" should be replaced with an "S" and so on.
We also call such a mapping a "dictionary", also "lookup table". Perm["A"] returns "S" given Perm = {"A":"S", "B":"Q", "C":"E"...} i.e square bracket notation (very common) accepts a key and returns a value, where a dictionary consists of key:value pairs.
We also call such a mapping a "dictionary", also "lookup table". Perm["A"] returns "S" given Perm = {"A":"S", "B":"Q", "C":"E"...} i.e square bracket notation (very common) accepts a key and returns a value, where a dictionary consists of key:value pairs.
All of the above is elementary in the sense of accessible without many prerequisites, an introduction to notation in connection with one of the most foundational of all data structures, the lookup table.
We want permutation dictionaries, any dictionaries, to start making their appearance early in one's career with notations, just as "data structures" more generally (ways of storing data) should be front and center in the early grades.
We want permutation dictionaries, any dictionaries, to start making their appearance early in one's career with notations, just as "data structures" more generally (ways of storing data) should be front and center in the early grades.
But then we see in many use cases we want to "apply" a data structure as a verb, back to group action.
We can chain the data structures together and have permutation follow permutation (composition, akin to "then"). This long "pipeline" of permutations may then be applied, as a chain, to any target string X. "THE RAIN IN SPAIN".
We can chain the data structures together and have permutation follow permutation (composition, akin to "then"). This long "pipeline" of permutations may then be applied, as a chain, to any target string X. "THE RAIN IN SPAIN".
What would "math" look like if the focus were "zero-sum" transactions and cycles of groups, instead of computing functions of variables and "solving" for inverses?Joe Austin
Permutations are simply re-arrangements and they may be strung together on a necklace (composed).
There's an identity permutation (every element maps to itself) and every permutation may be paired with its inverse, the one that goes the other way, thereby decrypting back to "THE RAIN IN SPAIN" from whatever scrambled string.
There's an identity permutation (every element maps to itself) and every permutation may be paired with its inverse, the one that goes the other way, thereby decrypting back to "THE RAIN IN SPAIN" from whatever scrambled string.
In object-oriented language we may wrap a dictionary inside a Perm type object which is also responsive to the "multiply" operator, repurposed to mean "composition". p = Perm(); q = Perm(); r = p * q.
Make two new perm objects and compose them: Perm() could start as random, given a default domain of objects to rearrange (e.g. A-Z and space character).
Make two new perm objects and compose them: Perm() could start as random, given a default domain of objects to rearrange (e.g. A-Z and space character).
In addition to chaining Perms, we might "call" them as in p("TEST PHRASE"), and getting back some new string. That's group action again. The set of permutations, closed, composable, associative, gets to "act on" target strings. That idea of group members as "verbs" with swappable "targets of operation" is pervasive.
Kirby
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