Dynamics on Graphs 28
8 views Jul 9, 2023
In the geometry of a linear space with a symmetric bilinear form, reflections in vectors and hyperplanes play a natural role. There is an important result here, which we call the Composite Reflection Theorem, which ought to be introduced in a second Linear Algebra course when studying such inner product spaces (or more general quadratic forms). It turns out that this theorem is intimately connected with the connection between root systems and Weyl groups and the subsequent geometry of Weyl chambers and hyperplane arrangements, and is applicable to general graphs.
This video is part of the Exceptional Structures in Maths and Physics: Dynamics on Graphs playlist.
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when we have an inner product or more generally a symmetric bilinear form the dot here representing
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that bilinear form okay so we have the vector W and here is the plane or rather the hyperplane
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of vectors which are perpendicular to W's or W perp okay all the vectors in this hyperplane
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are perpendicular W meaning that they're dot product with W is zero and now we're going to
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reflect this vector V in this hyperplane and we're going to denote that by R W perp V and
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the formula that we have for that is that it's V minus two times V dot w over w dot w
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times W this we recognize as a projection of V onto W so we're basically taking the projection
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and subtracting twice the projection to get over here Okay so it's an easy exercise to show that if
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you do this twice so if you square this operator you're getting the identity and the question we
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want to address today is the following if say Z is this reflection of V in W perp so here's V here's
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W here's the reflection Z of v in this hyperplane determined by W how do you express the reflection
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in the hyperplane perpendicular to Z in terms of the Reflections in the hyperplanes perpendicular
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to W and V so we're starting off with two vectors W and V they both have hyper planes we both have
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Reflections in those two hyperplanes so we have these two Reflections determined by V and W and
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now we're reflecting V in I'm playing on W again Z and we're interested in describing how this this
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new reflection is described in terms of the two original ones so let's have a look at this little
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diagram where I tried to sort of illustrate things I'm using the sort of standard Euclidean dot
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product in the plane here so that we can use our geometric intuition okay so there's W there's a
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hyperplane there H of W we're reflecting V in that hyper plane there's Z over there so the hyperplane
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corresponding to V is is this one here okay and the hyperplane corresponding to Z would be this
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one here so let's suppose that we take a vector U so that one there in blue and we reflect it in
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this hyperplane agent Z so it's R of Z perp of U there it is right there from here to here okay
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and I want to show you also over here from here to here this U prime here is the reflection of U in
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hyperplane H of W okay so from here to here and then if we take this U Prime and we reflect it
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in H of V okay so we go over here we get down this Vector this is U double Prime and then if
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we do one more reflection in H of W again okay then we get this Vector going to this
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Vector so this little diagram if you follow it suggests that the reflection determined by Z is
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alternatively the composite of three of the other ones namely an R W perp followed by an R V perp
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followed by our W perp right so it's actually a good exercise to draw this kind of thing and
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try it out for some other vector is U put the U in different places including along these significant
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directions here and see if if the pattern sort of maintains okay and here is what we're saying
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that if Z is equal to the reflection of vector V in the hyperplane determined by W
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okay then the reflection in Z perp is equal to the composite of the reflection in W perp
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production in V-perp the reflection in W perp usually when we write a composition like this
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we sort of read from the right to the left so the convention for compositions is that F Circle G of
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X is F of G of X so if we take X we apply G first and then we apply F so in this case it
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actually doesn't matter because the the left one and the right one are both the same okay
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but technically speaking what we're saying is if we have a vector U and we want to know what
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the left hand side evaluated at U is we take U and we apply this R W perpand then we apply
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this R V perp and then we apply this R W perp and that's how we get our Z perp
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now we're going to make a computation and it's a very pleasant computation but you
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know it does require sort of some sitting down and being patient and being careful and writing
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things out so if you're a student of mathematics this is a great exercise to put the video aside
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and just get a piece of paper maybe a couple of pieces of paper and work it out yourselves okay
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it will be good practice to hone your algebraic skills okay so we're taking it back to U and we
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want to figure out what is our Z perp of U and here's the definition it's U minus 2 U dot Z
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over Z dot Z times Z okay and what is Z well Z is is this thing here Z is the reflection
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of V in W verb so this is this is Z right here V minus 2 V dot W over W dot W times W
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that's it and there's another Z down here there's another Z down here they're done by themselves and
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over here there's Z so this is all our numbers this whole thing is a big number that's a vector
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there's a number times this vector we want to simplify this thing all right what happens
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well we sort of expand using the linearity here so here there's a um maybe take a plus
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sign we take minus two U dot V okay and then a minus times the minus is a plus so we're gonna
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get four uh U dot w okay this B dot w is just the scalar so it appears also there and the W
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dot W in the numerators it's just a scalar so it appears there when we distribute out
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the denominator we get a V Dot V we get minus four V Dot W squared over W dot W plus the square of
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the last thing so 4 V Dot W Squared and there's a W dot W but that'll cancel with one of the W dot
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W's on the bottom we just get a single W dot tell it down there and this thing here is just weird
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it was before okay what happens now is this bottom line here well you see um
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this thing here is V Dot V okay and um and this expression here and this expression
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exactly cancels so there's really only just a V Dot V in the bottom here so we're going to get
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minus two U dot V over V Dot B is the first term up here and then four well the V Dot V which is
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in the bottom will join the W dot W which is in the bottom so we're going to get in V Dot
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V W dot w in the bottom and a U dot W and a V dot W in the numerator and all multiplied
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by this Vector here so that's something of a simplification of the left hand side of
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our equation we've taken an arbitrary Vector U and applied reflection of that perp to it
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all right now the next thing is to work out this composition okay so here's the the composition
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here's our definition first we apply this right most operation and then the next one and then the
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next one okay so we start sort of with the inside this first one here this inside thing these R of
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W perp U that's U minus two U dot w over w dot w times W and we have to apply R V perp to that
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well that's going to be this thing by itself minus two times this thing times or dot with
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V over V Dot V times V and we have to take all of that and apply yet another R W perp to it
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but before we do that we're going to simplify a little bit so here's a U this thing here is this
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term here over here we have a 2 U dot V over V Dot V times V and a plus 4 U dot W [Music]
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W dot V W W V dot V times V okay now when we want to apply R W perp to this what do we get
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well we get this whole Vector there it is right there okay minus two times this whole Vector
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dotted with W over W dot W times W so we're using that same formula for the reflection in
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the hyperplane over and over again okay now what happens well so there's a U here okay there's a U
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and uh what else is there let's just span it out there so there's a U and and this term here
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U dot W W dot V there's something multiplying V okay this thing here okay is multiplying V
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there's a 2 U dot V over V Dot B there's that term there and this thing here is it
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expressed there this thing here is actually a multiple of W so we can sort of lump it together
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with this thing over here so um over here we have [Music] um there's minus 2 U dot W over W dot W
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but there's also another minus two of them here so all together minus four U dot W over W dot w
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and um and then there's uh this times this that's four U dot W over W dot W times the W
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and uh then this one here again the U dot V V dot W Ave V Dot V the W dot W this one down
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here and it's a coefficient of plus four and then this term times this is minus eight U dot
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W W dot V squared because there's another one coming here and a V Dot W W dot W squared in
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the bottom because there's another one down here all of that multiplied by W
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okay what is that well there's a U okay there's um we're then gonna write this one in terms of a
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common factor there's a minus two U dot V over V Dot V is a common factor between this thing here
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and also uh this one here this also has a two U dot V over V Dot V times another
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two V dot w over W W Times w so we have this thing times V that's expressed there
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um and this thing times W so oh these guys here are canceled okay that's an important thing to
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say and there's only really only two terms left here and one of them is is is this thing um times
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this one and the other one is this thing times this one so when we write it like this then we
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can scroll up and compare it with what we got over here and we see that it's happily the same thing
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so that's a bit of work right it's not obvious at all at least not obvious algebraically so
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it's a a reasonably you know subtle um sort of interesting theorem because the proof is
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is somewhat non-trivial it's a it's a somewhat you know involved calculation
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and let's just observe that algebraically these outside Reflections it might be useful to to think
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of this whole thing as a conjugation where we're multiplying by one thing on the on the left and
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the inverse uh on the other on the right side or this R W perp on the right and it's inverse
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on the left uh the reason for that is because these reflection operators when we Square them
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we get the identity so their inverses are the same as themselves but it's just consequence of
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that that we note that this kind of thing can be thought of as a conjugation so that this R Z perp
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is a conjugate of the R V perp and what we're conjugating by is the R W perp that's required
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to get from the V to the Z in the first place okay this is a really important theorem actually it's a
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an important geometrical theorem that in my view is such a central theorem that all undergraduates
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should see this theorem in a linear algebra course so if you're taking a linear algebra course
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especially maybe a second year a second version of it where you're delving a little bit deeper into
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the geometry Associated to symmetric bilinear forearms and quadratic forms then this is a
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really I think important result that needs to be identified and clearly stated and given a proof
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um just a small note that if V Dot V equals W dot W equals 2 okay then the argument
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simplifies a lot because well if you if you just go through the argument there's
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a lot of these W dot W's V Dot V's Etc in the denominator and if there are two then
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they just happen to cancel with the twos which are in the numerators so
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then you can go through and you'll see that the argument is is much simpler
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all right so now let's move to the vial group okay so we have x a graph this is actually quite
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General X is just a graph we have the space of populations on the graph P of X so now for any
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root population so any P which is a a root okay that is it's obtainable from a Singleton by a
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sequence of mutations let's call it P we're going to Define S of P to be the reflection
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in the perpendicular of P so we have the tits quadratic form that's that's the geometry so the
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vector P in the space determines a hyperplane and the reflection in that hyperplane given exactly
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by the same kind of formula we've been talking about we're going to now think of that reflection
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as being associated to this population p so now we have these Reflections not just for the Singleton
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populations the vertices themselves we now have a much wider family of reflections
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so suppose that P happens to be obtainable from a Singleton population say at X Delta
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X by applying a single mutation s of Y for some vertex Y okay so we start with the Singleton X
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and we perform a single mutation S sub Y to it to get this slightly bigger population P then
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this is exactly the kind of situation that we can apply this composite reflection theorem because
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we have already determined that this reflection in the vertex the S sub Y basic mutation it's
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actually a reflection with respect to the tis quadratic form it's actually R of Y perp
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okay so we're identifying y with the Delta function at Y the Singleton population
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okay so here's why here's some Vector X here's the hyperplane H sub Y which is the
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vectors which are perpendicular to Y and we're reflecting X in this hyperplane to get our
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Y perp of X and that's the population P okay but now from the composite reflection theorem that we
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have just figured out we can figure out what the reflection corresponding to P is the reflection
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corresponding to P or P perp which is not what we're calling s p is this composition
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our Y- perp followed by R X followed by R Y perp should have been R X perp I suppose okay
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and in terms of just ordinary mutations It's S of Y followed by S of X followed by S of Y
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so more generally if we have a general root population P which is obtainable from a
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Singleton Delta X by a sequence of mutations at various vertices Y1 Y2 up to Y k that's
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what a general root population looks like then correspondingly exactly the same argument is
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going to show that the reflection corresponding to P S of P which is the reflection in the plane
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hyperplane perpendicular to P that's going to be given by a similar kind of expression with an S
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of X in the middle and then on either side an S Y1 and then on either side of that an S of Y2 and
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then all the way up to S of Yk on either side because every time we perform another mutation
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the effect is to take the original thing that we have and sandwich it between two
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identical things the the the New Reflections of the things that we're adding to in the expression
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so this is a kind of a powerful thing because it gives us now control over
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what these Reflections s of P are doing we can see that for any root population p
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this reflection s of P that we've defined actually takes populations to populations
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it's not just a general reflection in the space it's actually preserving our set of populations
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why is that well we know that any mutation S of Y for Y any vertex of X sends populations
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to populations okay and since S of P is now a composition of mutations this is what this
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is saying S P is a composition of mutations at single vertices each one of those single
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mutations takes populations to populations so this big composite is also going to do the same
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so it's going to send populations to populations
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another consequence is that any root reflection S of P okay is actually conjugate in this vial
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group to a simple reflection or mutation Matrix S of X because if we take this formula that we
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have for S of P of x in the middle and S of Y1 that's Y one Etc then because all of these
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things here are mutations they're equal to their inverses so we can replace all the things on the
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left hand side with their inverses and once we've done that then we can write the whole
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thing as a conjugation because it's S of X times this product S Y1 S Y2 up to Y S Y k and over on
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this side it's inverse because when we take the inverse of this we get the inverse of the this
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S Y K on the left and then you know we change the orientation all the way up to the inverse of S of
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Y1 so this way we recognize that this S of P corresponding to the general root population
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that is conjugate to actually a simple
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reflection in a vertex so geometrically it's going to have a same kind of properties but
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that means that inside this Weyl Group which is typically very big we're getting a set of
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distinguished Reflections these s of P's which are indexed by positive roots because S of
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P is the same as S of minus P the reflection determined by a vector P and it's negative are
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going to be the same because the hyperplane that they generate is the same so we don't
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need to index over all root populations just over positive root populations okay so um for each one
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of these things we we get this reflection and so we have the sort of set of our Reflections inside
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the Weyl group they are sort of special special elements they're all of order two because they're
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Reflections and there's one for each positive root it's not a subgroup not not close but
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it's an interesting set and corresponding to this is a very fascinating geometrical
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configuration which is called a Hyperplane Arrangement okay so we're touching base now
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with another reasonably big subject in the overlap between combinatorics and geometry
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study of hyperplanes in in higher dimensional spaces where you hyperplane if you're in
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N-dimensional space a hyperplane is an (N minus one)-dimensional subspace so going through the
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origin and you can have a whole bunch of these things okay going all going through the origin
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and that configuration is something that comes up in a number of situations but in particular
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it's actually really important for us here because for each one of these positive roots
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actually for each one of their root vectors there's going to be a hyperplane
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of course for a vector and it's negative the hyperplanes are going to be the same but these
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hyperplanes are exactly in correspondence with the elements of T because the elements of T are just
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the Reflections in these various hyperplanes so instead of thinking about the positive Roots as
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a set of vectors another way of thinking about what we have here is a collection of hyperplanes
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in this big ambient space of populations
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and uh just as an example to finish off for the case of A2 favorite example there are three
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positive Roots so here's the way we like to think about things maybe a very Square lattice kind of
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fashion okay here in red we see the three positive Roots say Delta one Delta two and Delta one plus
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Delta 2 and down here on the opposite side here are there negative so there's six with these
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roots all together and the three hyperplanes determined by these things so the hyperplane
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things core corresponding to Delta one things that are perpendicular to other one that's that blue
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line there the thing is um which are perpendicular to Delta Two it's that blue line there
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and the things that are perpendicular Delta one plus Delta two that's that blue line there
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so this is the hyperplane arrangement that corresponds or that we find corresponding to this
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family T of Reflections in the VA group for the case of A2
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it's a very interesting story so the geometry is getting very rich and the algebra is really rich
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and um and this thing combines to be hugely important in in mathematics and in physics