LINEAR ALGEBRAIC GROUPS: LECTURE 6

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1 LINEAR ALGEBRAIC GROUPS: LECTURE 6 JOHN SIMANYI Grassmaias over Fiite Fields As see i the Fao plae, fiite fields create geometries that are uite differet from our more commo R or C based geometries These ted to be coected to -deformed versios of itegers, factorials, biomial coefficiets ad other uatities familiar from combiatorics Theorem Over the field F, Gr, To prove this, we reuire a few lemmas Lemma -Pascal Lemma For ay,, + is the oly fuctio of such that: Proof Of Lemma It is clear that rules ad uiuely determie a fuctio, ust as the usual Pascal triagle built o -choose- does Recall that [] [] [ ] Date: October, 6

2 It is clear that have LINEAR ALGEBRAIC GROUPS: LECTURE 6 obeys reuiremet To show it obeys reuiremet, we simply compute We + [ ] [] [ ] + [ ] [ ] [ ] [ ] [ ] + [ ] [] [] [ ] [ ] [] [ ] [ ] [] [ ] [] [] [ ] As metioed i the proof, this leads to a aalog of Pascal s triagle called the -Pascal triagle: 3 Numerically, we ca picture the diagoals as actig almost like the usual Pascal triagle - we add two o oe row to fid the value cetered betwee them o the row below However, the -Pascal triagle icludes multiplyig by a power of o the left diagoals, based o its positio i the triagle

3 LINEAR ALGEBRAIC GROUPS: LECTURE 6 3 Here s the triagle built to the fourth layer: I this case, the value 4 [] To prove our mai theorem, we reuire oe more lemma is the first that does t look like a -iteger, or somethig of the form Lemma Grassmaia-Pascal Lemma over F Let F be a fiite field The Gr, Gr,, Gr, Gr, + Gr, Proof Of Lemma is clear, as there is oly oe zero-dimesioal or -dimesioal subspace i k for ay field k For, let be give, ad choose ay -dimesioal subspace i k Let s call it k, ad use the atural embeddig coordiates k {x, x,, x, k } Suppose L Gr, There are two choices: either L k, or it is ot If L k, the L Gr, If L k, the it has exactly oe basis elemet ot i k, so I a proective sese, we ca write dimk L L L k + x, x,, x,, where the geeratig vector is uiue up to addig vectors i L k By dimesioality argumets, we have that k L k I the field F, a subspace of dimesio has elemets Together, there are Gr, possibilities that lie i k, ad Gr, that do ot The sum i the lemma follows Fially, the theorem follows from the two lemmas

4 LINEAR ALGEBRAIC GROUPS: LECTURE 6 4 Pascal s Triagle ad Bruhat Cells The previous sectio focused o fiite fields However, there is a versio of Grassmaia-Pascal lemma that applies to arbitrary fields: Theorem Let k be ay field, ad, ay atural umbers with The there exists a biectio of sets Gr, Gr, + k Gr, Here + meas disoit uio, ad is Cartesia product The proof is idetical to the Grassmaia-Pascal lemma This result ca be used to decompose ay Grassmaia ito a disoit uio of copies of k i, which are called Bruhat cells for the Grassmaia Example 3 Recall that ad cosider Gr4, We ca decompose this as Gr, Gr, k, Gr4, Gr3, + k Gr3, Gr, + k Gr, + k Gr, + k Gr, k + k Gr, + k Gr, + k Gr, + k Gr, + k Gr, k + k k + k k + k k + k k + k k k + k + k + k + k 3 + k 4 Writte this way, we ca see -Pascal s triagle showig up Each cell arises from a uiue dowward path from the tip of the triagle to the particular -biomial coefficiet Reflectig a bit o this, we have Theorem The umber of Bruhat cells i Gr, is the umber of distict paths from the vertex of the -Pascal triagle to the poit, The umber of such paths is the ordiary biomial coefficiet Note that steps to the right ad dow do t multiply they are arrows, while those to the left ad dow do multiply by i, for a particular i Here are all six distict paths for Gr4, :, : 4, :

5 LINEAR ALGEBRAIC GROUPS: LECTURE 6 5 If you compare this to the earlier picture of the -Pascal triagle, you ca see how each power of arises i that triagle If we reflect o the proof of the Grassmaia-Pascal lemma, the idea becomes clear Suppose we take a -dimesioal subspace L k As we cosider the itersectios L k, L k, L k,, we see that at each stage either the dimesio stays the same or icreases by oe We ca keep track of this usig a path from the top of the -Pascal triagle dow to its, etry Each time the dimesio stays the same there is o choice ivolved, ad we go right ad dow, ad Each time the dimesio icreases by oe, there is a choice of how L k i could be exteded to a subspace of k i+ havig oe extra dimesio, ad we go left ad dow If we cout all choices we make o this trip, we get the cardiality of the Bruhat cell cotaiig L This is d, where d is the dimesio of that Bruhat cell As a exercise, prove that the dimesio d ca computed i the followig easy way Each Bruhat cell i Gr, correspods to a path from the top of Pascal s triagle to the, etry To compute the dimesio d of this cell, take the rectagle formed with vertices,,,,, ad,, ad cout the suares to the left of the path Here are a few examples from the decompositio of Gr5, 3 to show the process: 3 4 6

Math 155 (Lecture 3)

Math 155 (Lecture 3) Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,