Positive roots for ADE graphs (with exception E8)
Wild Egg Maths
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The mutation game approach to Coxeter Dynkin graphs gives us a powerful simplified way of describing and understanding classical root systems associated to the ADE graphs. We show directly using a case by case analysis that each of these graphs does indeed have a finite set of roots, and that the positive roots can be organized into a poset. This completes our main classification theorem.
These posets of positive roots are remarkable combinatorial objects that have many applications in mathematics and physics. However for E8 we still need to exhibit this poset a nice challenge for viewers!
Errata note (thanks to Christiaan Mantz!): At around 29:07 in the E7 positive root poset there is an edge missing from {{1}, {001000}} to {{1}, {011000}} i.e. between the 3rd entry in the 2nd row, and the 2nd entry in the third row.
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hello everyone i'm norman walberger and
00:03
thanks for joining me on this adventure
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towards understanding dynamics on graphs
00:07
today we're going to introduce positive
00:09
roots for ade graphs
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this is completing this
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initial goal that we have of
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classifying those graphs that have a
00:19
finite
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set of roots
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so
00:23
i'm going to review some of the basic
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terminology and the starting point and
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then we're going to
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look carefully at these positive routes
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for the ad graphs and convince ourselves
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that they actually do form finite
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collections
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and we're going to have to look rather
00:40
carefully
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do some bookkeeping
00:43
make some diagrams and sort of check
00:45
things by hand so it's a pretty hands-on
00:49
discussion today and later on we'll see
00:51
something a little bit more abstract
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okay so the general setup is we're
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starting with a simple graph x okay
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and
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we'll let p of x denote the populations
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on x that is the integer valued
01:04
functions on the vertices of x
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okay at some point we'll assume that x
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is connected but actually to begin with
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that's not really necessary
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we're thinking of p of x as a like a
01:14
vector space or a linear space but the
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coefficients are
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integers so this is a space over the
01:20
integers okay but we can add functions
01:24
we can scale or multiply functions so
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it's really just like a vector space
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okay now for a vertex x
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let's let n
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of x denote the set of neighbors of x in
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other words those vertices which are
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directly adjacent x v and h and now the
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crucial definition we had was of a
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mutation so if x is a vertex we're going
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to define the linear transformation s
01:47
sub x so that's a
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map from populations to populations so i
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have to tell you that if some population
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is called p
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then what is the effect of this mutation
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on p so i have to define what is p s of
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x that's the notation we'll use the
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population p
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followed by the mutation s sub x
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okay so this is going to be a new
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population which means it's a function
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on vertices and here's the formula for
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it okay so if y is now an arbitrary
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vertex of x
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then p
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s sub x of y is what
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two cases
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if
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the y the input is actually x
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then
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the number is
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minus p of x plus the sum
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over all z which are neighbors of x of p
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and z
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in other words when we mutate
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s sub x the value at x
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changes as follows it negates and then
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you add the
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populations of the directly adjacent
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neighbors
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and otherwise in other words if you're
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looking at some other vertex other than
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x the population is unchanged
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okay so that's the crucial notion of
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mutation now let's let delta x denote a
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singleton population
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that means it's zero everywhere except
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that one vertex
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the vertex x and at that vertex it's
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equal to
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one okay and then we defined a root
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population to be a population which can
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be obtained by starting from some
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singleton it doesn't matter which
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singleton you just choose a singleton
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vertex somewhere
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and then you apply arbitrary mutations
03:30
to it so anything that you can get from
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a singleton population is what we call a
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root population
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and we'll let r of x denote the
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type of root populations
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i'm not going to say set so quickly
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because
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this is not always
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a legitimate set in the sense of a
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finite set sometimes it has an unbounded
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aspect and in fact that's exactly what
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we're interested in under what
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conditions do we get a finite set okay
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so we have determined that if x is
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connected and it has a finite number of
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root populations then x must be one of
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these aede graphs today we want to
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convince ourselves that the converse is
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also true that if x is an ade graph
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then r of x is finite
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and we'll be
04:17
able almost to do this but there will be
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one exception that looms sort of outside
04:22
our view because it's a little bit big
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that's going to be e8
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okay so right now our proof of this will
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be
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computational but
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later on we're going to see a more slick
04:34
geometric proof which is a lot harder to
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to come by if you're not sort of already
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familiar with the situation
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okay but what we're going to see today
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and it's so important is that we're
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going to see these
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these structures emerge which are called
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classical root systems okay so these are
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examples of exceptional structures that
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play a big role in both mathematics and
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physics
05:00
classical root systems okay
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the root systems are just these sets of
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roots for for graphs for the ade graphs
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but because we're looking at it from
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this mutation point of view we're going
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to get a
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new understanding or sort of a new
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realization of these classical
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root systems okay so what we're doing
05:25
here is a foundation for a lot of
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additional things that we will be doing
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so i want you to have a really careful
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and close look at these structures that
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we're going to be looking at today
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okay are we ready oh first before we get
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on to that i have to make an important
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two observations the first is that if
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you have a vertex x
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and
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you look at the singleton at x and you
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apply the mutation s sub x to that
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so you have a singleton population at x
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there's a one at x zero as everywhere
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else and you mutate at x then what's
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going to happen
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well the one is going to change to a
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minus one and then you have to add the
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sum of the neighboring values but the
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neighboring values are all zero so the
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one is going to turn into a minus one
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and otherwise it's going to be exactly
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the same
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so
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the effect of the mutation at x on the
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singleton x is to negate it
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now this is a very
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easy but important observation because
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it means that
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the entire
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story of roots is symmetrical under
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interchange of sine because you see now
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you have these two roots there's
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delta x and
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minus delta x they are both roots now if
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we apply any mutations to x
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any sequence of mutations tags to create
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some more general root if we
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apply exactly those same mutations to
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minus
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delta x we're going to get exactly the
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negative of what we got up here
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okay so that's allowing us to deduce
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this second corollary that if p is a
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root population then so is minus p
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okay
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that means that these these roots are
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going to be sort of divided into into
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two halves there's the halves that you
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can get by starting with the delta x and
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doing mutations on that and then the
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other half that starts with minus delta
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x and doing mutations on that
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and one way of agreeing to keep them
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separate well i'm going to write that
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down to keep them separate
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let's agree
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not
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capital n not to
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mutate
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uh say delta
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sub x at
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the vertex
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x
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in other words
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we're going to be considering all the
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possible things that we can get
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from delta x by applying mutations but
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not allowing ourselves to mutate at a
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singleton
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okay so we are we're going to avoid i.e
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we avoid
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mutating
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singletons
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to get
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their negatives because we know exactly
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what's going to happen when we do that
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so we want to find out what happens when
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we don't do that so we're
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restricting ourselves to just half of
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the story it means that we don't have to
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illustrate as many roots we're only
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illustrating half of them
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okay so
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let me illustrate what's going to happen
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okay so let me illustrate what's going
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to happen i'm going to start with
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an a n graph and i'm going to do a4 and
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you'll see that once you understand a4
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you can pretty well understand the
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general a n story so here's a4 okay it's
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a graph with vertices that we're going
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to call one two three and four in that
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order
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and
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this here is a picture of the positive
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roots that is
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those roots which are obtained from
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singletons by not
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mutating at the singleton at that vertex
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and that was by not applying s sub x to
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delta sub x
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so
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up here at the top so we read from top
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to bottom at the top here you see the
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four singletons and these are
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represented as functions on the vertices
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in the obvious way so this element here
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this number one zero zero zero that's a
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population that's a singleton population
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which is just delta sub one it's got a
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one at
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one and zero's everywhere else
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here's the delta at two here's the delta
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at three here's the delta at 4.
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and now what i'm going to exhibit here
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is all the additional
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roots that we can get by mutating the
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these original delta ones
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okay so for example if we start with
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this one here this is delta sub one and
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we apply the mutation at two so it's a
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one here and we're going to mutate here
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what's going to happen well the zero is
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going to go to minus zero and then we
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have to add one so we're gonna get a
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population which has a one here and a 1
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here that's exactly this one here
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now this one here however could also be
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obtained in a different way it could be
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obtained by starting with the singleton
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at 2 and performing a
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an s sub one mutation in other words
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mutating at one okay and that's what
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these bonds represent so these bonds are
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the ways of getting this one here either
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from this singleton or from the
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singleton this represents a mutation
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this represents a different mutation
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and similarly we could get the
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population which has values one at two
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and three either from the singleton at
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two or the singleton at three and we
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could get the mu
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and we get the population that has ones
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at three and four
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either from the singleton at three or
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the singleton and four
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okay so it's kind of a next layer level
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down the population has increased okay
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now we have populations of two a total
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of two
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but we can keep on applying a mutation
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so if we apply um another mutation to
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this one say we apply s sub three to
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this well it's going to add an extra one
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at three and we're going to get this one
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or if we take say this one here and we
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apply an s4 then we're going to get this
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one and again you could get this one
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also by right here and then once we have
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these two then we could apply one more
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and we could get
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this 1 1 1 1.
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and now we can
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see that these are the only
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roots that we're going to get
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that are positive in other words that
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are not obtained from this from this
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symmetry by negating a singleton at the
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beginning
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and how do we check that so we check
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that by
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taking one of these and asking all right
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if we look at any possible mutation that
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we could do this one let's look at this
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one okay if we do
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a one mutation then that'll send us to
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this one
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if we do a two mutation okay that will
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send us a two mutation that's that one
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there
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that will send us
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it will kill that one there and it will
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send us over over to here
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if we do a three mutation
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that's this one here then what's the
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effect of that then one goes to minus
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one and then we have to add a total of
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two that leaves it unchanged so a three
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mutation will leave this unchanged now
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what about a 4 mutation that will kill
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that final one there and send it over
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here so each of the four possible
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mutations that we could do
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is going to take this thing here to one
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of the existing
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fellows on our
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diagram
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and we can check that for each one of
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these you can check that for each one of
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these things the same is true you apply
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any one of the four mutations to this
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one and you can get one of its neighbors
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here or itself
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that is then a justification for saying
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that these are all the positive roots
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now these are not all the roots okay
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these are the positive roots which we
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denote with a plus sign here so these
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are the the roots obtained from
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singletons by agreeing not to do this um
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kind of negating mutation at the
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beginning if we agreed to do that as
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well then we would get another
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additional 10 roots which would be
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exactly the negatives of these ones
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okay so all together we can count how
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many we have here there's sort of four
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in this top
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layer level and then there's three and
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then there's two and then there's one a
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total of ten
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so i might also add so altogether
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r of a sub four
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the size of that
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will be 2 times 10
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which is 20.
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so all together there are 20
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roots for the graph a4
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and it's not hard to generalize this to
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a n as i've done down here so
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if you have a look at this formula here
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uh the same kind of pattern is going to
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happen this um
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this sort of layering as we increase the
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population you're going to get um
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more and more
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uh strings of ones with zeros on either
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end in general okay and the numbers are
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going to decrease linearly so starting
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with n at the top and then n minus 1 and
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then n minus 2 all the way down to 1. so
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the total number of positive roots for a
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sub n
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after you think about it for a little
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while and realize that the same
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phenomenon is going to happen will be
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the sum of these things which of course
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is n times n plus 1 over 2.
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and so i might say so also so also the
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number of
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just ordinary roots of a sub n is going
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to be twice that will be n times n plus
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one
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okay let's look at d4 okay d4 is also a
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graph with four elements it's also a
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tree but it has this form here
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and i'm going to call the vertices one
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two three and the zero is going to be up
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here
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like a t upside down t shape
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and a population on this graph is going
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to be indicated by
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an expression like this you can see the
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three
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values at the bottom and then the ones
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up the top that corresponds to the value
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at zero
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and here up at the top we see the four
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singletons singleton at one singleton at
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zero singleton at two singleton at three
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and now
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we see a similar um story we just see
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the the effects of mutations on these
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singletons and how we can generate
16:12
increasing
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populations so let's start on this side
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for something different so we're
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starting with a delta function at 3 okay
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if we apply
16:22
s sub 2 we're going to get this one here
16:25
and then we could apply either s sub 0
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in which case we would go here or we
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could apply s sub 1 in which case we
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would go over here
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having done that then we could apply the
16:37
other one so if we've just applied
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s sub 1 then we can apply s sub 0 when
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we get this one here
16:43
so now we also have a bunch of ones
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but they're in a little bit different
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configuration than last time and now
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there's one more that we can do which is
16:52
the s sub 2.
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that will take that middle one there and
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change it to minus one plus three which
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is two
17:00
so we get this one here which is one one
17:02
two one that population is the maximum
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population obtainable on d4
17:09
and you can check that how by just
17:12
looking at all of these other ones and
17:14
asking well you know let's say this one
17:16
here what are all
17:17
the
17:18
additional
17:20
populations i can get by starting with
17:22
this one and performing one of these
17:24
four mutations
17:26
okay i can get either this one or this
17:29
one or this one or this one i stay
17:31
inside this set
17:33
what about this one here what happens if
17:35
we do more on this one well if we do a
17:37
an s2 we're going to get back to here
17:39
what happens if we do an s1
17:42
well the 1 changes to -1 then we have to
17:44
add 2 which
17:45
brings us back to one so this thing is
17:48
not changed under s1
17:51
so it's still the story that this set is
17:55
closed under
17:56
these um
17:58
mutations
17:59
taking into account that if you actually
18:01
take one of these and apply
18:04
the mutation at that vertex well then
18:06
you're going to get to its negative so
18:07
you then you're going to get to sort of
18:08
the mirror image of this collection
18:10
which is a
18:12
whole bunch of things like this but just
18:14
with the negative values of these ones
18:18
all right let's count all the positive
18:19
roots uh there's four in this layer
18:21
three in this layer three in this layer
18:23
one this layer one's layer for a total
18:24
of 12. so there's 12 positive roots and
18:26
so 24 roots all together
18:31
okay so we're verifying that this graph
18:33
here has a finite set of roots and we're
18:36
doing it explicitly
18:38
combinatorialists are like a bookkeeping
18:40
we're just checking looking at all the
18:42
possible things that we can get and just
18:44
checking that we can't get any
18:45
additional ones
18:48
it's naive but it's a really good thing
18:50
to do because it will give you some
18:52
familiarity
18:54
with these um these sets of positive
18:57
roots which
18:59
are going to be really important for us
19:01
okay these are
19:02
absolutely
19:04
fascinating
19:06
structures containing really interesting
19:08
geometry and really interesting
19:10
combinators and lots of connections with
19:12
algebra of all kinds
19:14
and physics too
19:16
okay so what about generalizing so let's
19:18
go to d the more general d graph d4 is
19:21
always a little bit special here is d5
19:25
okay
19:26
verses one two three four zero up here
19:29
and then here is the picture
19:33
there it is there okay
19:35
there's the singletons
19:37
so the
19:38
population
19:40
total two population total three four
19:44
five there's sort of two different ones
19:46
there's one that looks like the one that
19:48
we've already gotten but with an extra
19:50
zero and then we can go to six and then
19:52
we can actually go the maximum one is
19:54
this one seven that looks like what well
19:57
it has ones on the on the end points
20:00
the extremities and it has a string of
20:03
twos in the
20:04
middle and that's indicative of the
20:07
general story
20:08
uh if you look at d sub n for a larger n
20:12
it's going to be like this it's going to
20:13
be more complicated but ultimately
20:15
you're going to end up with a maximum
20:16
population which has ones on the on the
20:18
outside and twos along this this middle
20:22
string
20:24
you might like to check that and you
20:26
might actually uh here's a good question
20:28
for you how many positive roots are
20:29
there for general dn how many were there
20:31
for this
20:33
five
20:34
four four
20:36
three
20:37
two one one
20:39
twenty of them
20:42
by the way
20:43
while we're here let me let me talk a
20:45
little bit about this structure here so
20:47
this is
20:48
what in common torx is called a posit
20:51
it's a partially ordered set
20:53
okay
20:54
and
20:57
so
20:58
this is intimately connected with this
21:00
branch of combinatorics having to do
21:02
with ordering so posted theory is very
21:04
rich and this is actually a really nice
21:06
post set uh from many points of view we
21:09
certainly see
21:10
that um
21:12
you know it's
21:13
it's layered
21:15
in combinatorics they say graded this is
21:17
a graded postset what does that mean it
21:19
means that we can split it into layers
21:21
just like you can see visibly see here
21:24
so that the covering relations of the
21:26
postset are between adjacent levels what
21:29
does that mean so it means that
21:31
you know if we agree that
21:33
that um that one of these is
21:36
is related to another one which can be
21:38
obtained by a sequence of mutations
21:41
okay
21:42
um where where you're increasing in in
21:45
in the population so you're increasing
21:47
the population with this mutation
21:48
increasing the population with this
21:49
position increasing the population of
21:51
this mutation so we would say that so
21:53
this one here sort of dominates this one
21:55
because it's got a bigger population
21:57
which is obtainable from the original
22:00
one
22:01
however say if you compare this one here
22:03
with this one here they are not directly
22:06
comparable because this one has a bigger
22:08
population but you can't get to this one
22:10
from here by a bunch of
22:12
increasing populations so as far as the
22:15
partial ordered set goes these are not
22:18
comparable
22:19
okay so these covering relations are
22:21
kind of the minimal
22:23
um
22:24
the minimal
22:27
examples of
22:28
something being bigger than something
22:30
else okay so this one is bigger than
22:32
this one but there's nothing in between
22:34
them so there's a covering relation
22:35
which is illustrated with these little
22:37
edges that we've drawn this is called
22:40
the hasa diagram of this post set
22:42
and you know we're seeing that this has
22:44
a graded um structure so it's a graded
22:47
post-it that's pretty interesting
22:49
and also it has a kind of um
22:52
well there's a
22:54
nice sort of
22:56
sequential aspect to the the number of
22:59
elements in each layer right it's going
23:01
down weekly five four four three two one
23:05
one it's not going up and down it's kind
23:06
of going in the same direction
23:09
there's something that's called the
23:09
whitney numbers of uh of a post set okay
23:12
so it's a it's a somewhat uh sort of
23:15
interesting and special property of of
23:18
this post that it's graded and that
23:20
these whitney numbers are are weakly
23:22
decreasing
23:25
so natural question is as we investigate
23:27
uh additional graphs you know is this
23:29
likely to still be happening or is this
23:32
just an accident because this is a
23:33
relatively small
23:36
graph
23:37
well let's go to the next one e6 so this
23:40
is one of our exceptions our first
23:42
exceptional
23:44
one c6 e7 and e8 so hear the story um it
23:48
gets surprisingly kind of interesting
23:51
and complicated so here's e6 versus one
23:54
two three four five zero okay we are
23:57
going to be talking a lot about e6 e6 is
23:59
a really really interesting thing
24:01
connects with all kinds of interesting
24:02
things between the 27 lines on a cubic
24:05
all kinds of classical geometry um it's
24:07
it's it's mind-boggling but this is a
24:10
really great starting point you know
24:12
that
24:13
we can generate this structure
24:16
of this root system for e6 in this
24:19
elementary way by playing the mutation
24:21
game and we get a realization of it
24:23
which is quite different from that that
24:25
you find in the classical texts on the
24:27
subject
24:28
so you can look up classical text bor
24:30
bache many many texts on lee theory will
24:34
have descriptions of the root systems of
24:36
e6
24:37
and
24:38
you're unlikely to to find a picture
24:40
like this i think but
24:42
it's a big subject i could be wrong
24:45
okay so let's have a look at this what
24:47
do we get well there's these six
24:50
singletons up at the top
24:52
and this one here is a little bit
24:53
different right that's that one right
24:54
there it's sort of squeezed in in
24:56
between
24:57
these ones here so it's a little bit
24:59
arbitrary where i've put it
25:03
okay and then and then you know i want
25:05
you to sort of stare at this and check
25:07
that i've kind of done this right that
25:10
you know there are bonds between
25:12
things representing mutations that
25:14
actually go from one
25:16
root to another
25:18
increasing as we go down increasing in
25:20
population as we go down
25:22
so this one here for example could be
25:24
obtained from this one or from this one
25:26
but not from this one
25:28
okay
25:31
okay so there's so then five at this
25:32
level and then five this level five at
25:34
this level
25:35
and then we go down to some fours it's a
25:38
little bit asymmetrical you know there's
25:40
some asymmetry here because of this one
25:42
here it's a little bit
25:45
a little bit asymmetrical at least the
25:46
way we're representing and then if we
25:48
keep on going down there's four and then
25:50
three and then three and let's just
25:52
scroll down here there's the three and
25:53
three and then there's two and then
25:55
there's one and then there's one and
25:56
then there's one
25:58
and the maximum population which is
26:00
particularly interesting for us is this
26:03
population that has one at the these um
26:07
sort of horizontal endpoints and goes
26:09
one two three two one along the bottom
26:12
and
26:13
in the zero spot it's two
26:16
that's the maximum population
26:19
okay so that one has the property that
26:20
if you if you hit it with a general
26:22
mutation most of the time it stays where
26:25
it is
26:26
all right so if i mutate this say at
26:29
that position there where it gives us
26:30
the position four i get minus two plus a
26:33
total of four that's two it doesn't
26:34
change
26:37
um if i even if i mutate here where the
26:40
three is the three goes to minus three
26:41
and then i get a total of six that
26:43
doesn't change
26:44
the only mutation that's going to affect
26:47
this thing is the mutation at the vertex
26:49
zero which is going to send it back
26:50
along this edge to this one here
26:55
so again if we want to convince
26:57
ourselves that these are the only
26:59
positive roots and we have to have some
27:01
kind of check you know so we have to
27:03
have a look at anyone like this and say
27:05
okay if i apply any mutation to this one
27:07
am i going to stay in this set
27:11
okay so it's a check it might take you
27:13
you know
27:15
15 minutes to do
27:16
it's not gonna take you that long but
27:18
you know it'll it's something to do
27:20
and what's the total number so we add up
27:22
all the um
27:24
the graded levels six five five five
27:28
plus four there's one four and then a
27:31
couple of threes then there was a 2.
27:34
just check that
27:35
ah there was a 2 yes there and then
27:37
there were 3 ones for a total of 36
27:40
so
27:42
the total number of
27:44
positive roots is 36 of course the total
27:46
number of roots will be twice that
27:47
because the negative of any one of these
27:49
is still a root so that'll be a 72.
27:56
for those of you who know some lee
27:57
theory
27:59
there is a simple lead algebra
28:01
associated to e6 okay and you may know
28:04
its dimension is 78
28:06
okay the 78 is coming from the 72 that
28:09
i've just told you about plus the six
28:12
here okay
28:14
four reasons that we will describe later
28:16
on
28:17
all right so here is e7 it's pretty
28:19
interesting it's already pretty
28:21
complicated let me scroll here so you
28:23
can see the whole thing
28:24
whoops
28:27
let's go back
28:30
uh let me do that again a bit slower
28:31
okay you can see there now down there's
28:33
four and then there's twos and then
28:35
there's some threes and there's some
28:37
fours
28:38
and fives and so on
28:41
and it's
28:42
it's pretty interesting
28:44
um
28:46
and you could check that these are all
28:48
the positive roots
28:50
by checking for each one of these things
28:52
that any one of these mutations any one
28:54
of these seven possible mutations is
28:56
going to keep you inside this set
28:59
the only exception to that is if i
29:01
choose one of the original singletons
29:03
and apply a mutation at that
29:06
point then i get its negative i get to
29:08
the
29:09
mirror image on the negative side of
29:11
everything
29:14
and what's our grand total our grand
29:16
total is 63.
29:18
so
29:20
the total number of positive roots is
29:22
twice that
29:23
which is
29:24
120 six
29:27
and the dimension of the corresponding
29:29
lia algebra you have to add
29:31
seven to that to one twenty six plus
29:33
seven is 133 another famous number
29:38
okay so there's um lots to do and and
29:41
these are
29:42
really interesting
29:44
combinatorial structures we're seeing
29:46
that same phenomena
29:48
that we observed before this is a graded
29:50
post set
29:51
and the
29:53
sizes of the gratings are weakly
29:55
decreasing so it kind of has a you know
29:58
a spindle kind of shape or half the
29:59
spindle shape
30:01
and um and the other interesting thing
30:03
of course is this maximum
30:06
the maximum population
30:10
that's got population one at the right
30:13
then two three f to a maximum four in
30:16
that that node that note of degree three
30:18
and then three and two on the other side
30:20
and above it too
30:24
so those are all the ade
30:27
graphs i mean the a's and the ds are two
30:29
infinite families but we've seen the
30:31
pattern there and then we have e6 and e7
30:33
but they're still e8 okay and e8 is
30:38
like e7 but it's more complicated
30:41
so this is a great challenge for you
30:43
okay great challenge see if you can
30:45
create the post set of positive roots
30:48
for e8 we will be coming back to this at
30:51
some point because e8 is of fundamental
30:54
importance in mathematics and physics
30:56
these days also
30:58
really interesting and this is giving us
31:00
a new point of view on the positive
31:02
roots
31:04
the root system
31:06
of e8 which supports all kinds of
31:09
interesting
31:10
geometry
31:12
and
31:14
and combinatorics and associated algebra