A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

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1 A Pieri-type formula and a factorization formula for K-k-Schur functions Motoki TAKIGIKU The University of Tokyo February 22, 2018 A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

2 Introduction K-k-Schur functions g (k) :symmetric functions parametrized by k-bounded partitions Roughly speaking... they appear in type affine A combinatorics (A) partitions P, Schur functions s, symmetric groups S n, Grassmannian,... (affine A) k-bounded partitions P k, k + 1-core partitions C k+1, affine symmetric groups S k+1, affine Grassmannian, k-schur functions s (k), K-k-Schur functions g (k),... A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

3 Introduction History (conjecturally) equivalent definitions of the k-schur function s (k) : Using tableaux atoms (Lascoux, Lapointe and Morse, 2003) Using a certain symmetric function operator (Lapointe and Morse, 2003) Using a Pieri-type formula (Lapointe and Morse, 2007) shown to be the Schubert basis of the homology of the affine Grassmannian (Lam, 2008) K-theoretic version: K-k-Schur function g (k) as the Schubert basis of the K-homology of the affine Grassmannian (Lam, Schilling and Shimozono, 2010) Pieri-type formula (Morse, 2012) we start here As a generating function of strong marked tableaux (Lam, Lapointe, Morse and Shimozono, 2010) A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

4 Introduction Outline Start with the Pieri formula for K-k-Schur functions g (k) Consider g (k) := g µ (k) µ Prove a Pieri-type formula g (k) h r = µ ± g (k) µ where h r = h 0 + h h r. Prove a k-rectangle factorization formula g (k) R t (k) = g R t g (k). (R t = (t k+1 t ) = (t,..., t)) Main tool: Bruhat orderings of affine symmetric groups A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

5 Background Objects 1 Background Objects 2 Weak strips and (K-)k-Schur functions 3 Results A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

6 Background Objects Preliminaries: Type A P = { = ( ) i Z 0, i i < }. e.g. (4, 3, 1) its Young diagram (French notation) Λ = Z[h 1, h 2,... ]: the ring of symmectic functions, where h r = x i1... x ir. i 1 i r Schur function s ( P) is characterized by the Pieri rule: h r s = s µ for P, r 1 µ/:horizontal strip of size r (µ/: horizontal strip µ 1 1 µ ) A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

7 Background Objects Preliminaries In the theory of (K-)k-Schur functions, Underlying objects: P P k ( C k+1 S k+1 ) Pieri rule : horizontal strips weak strips A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

8 Background Objects Fix k Z >0. P k = {k-bounded partitions} := { P 1 k} C k+1 = {k + 1-core partitions} := {κ P No hook with length = k + 1} s 2 i = 1 S k+1 = s 0, s 1,..., s k / s i s i+1 s i = s i+1 s i s i+1 s i s j = s j s i (i j 0, ±1) (all indices are mod k + 1) S k+1 = s 1,..., s k S k+1. S k+1 = {affine Grassmannian elements} := minimal length representatives of S k+1 /S k+1 = {w S k+1 l(wx) = l(w) + l(x) ( x S k+1 )} = {w S k+1 any reduced expression for w ends in s 0 }. Theorem P k C k+1 S k+1. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

9 Background Objects Bijection P k C k+1 S k+1 p: C k+1 P k ; κ given by i = #{j (i, j) κ, hook (i,j) (κ) k}. Example. k = 4 (C 5 P 4 ) p count the cells with hook length k A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

10 Background Objects c: P k C k+1 ; κ Example. k = 4 (P 3 C 4 ) c() 1: for i = l(),..., 1 : 2: while there is a cell with hook length > k in i-th row : 3: slide i-th row to the right A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

11 Background Objects s : S k+1 C k+1 Def. residue of a cell (i, j) := j i mod k + 1 Z k+1. Action of s i on C k+1 : (exactly one of the following happens) if there exist addable corners of residue i, add them all if there exist removable corners of residue i, remove them all otherwise, do nothing Example. k = s 2 s s 3 s s 1 Fact This gives a well-defined action of S k+1 on C k+1, and induces a bijection s : S k+1 C k+1; w w. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

12 Background Objects P k S k+1 ; w reading the residues from the shortest row to the largest, and within each row from right to left. Example. k = 3 (C 4 P 3 S 4 ) s c() w A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

13 Background Objects The strong and weak orders (W, S): a Coxeter group The (left) weak order L is given by the covering relation u < Lv l(u) + 1 = l(v), su = v ( s S). The strong order (Bruhat order) is given by the covering relation u < v l(u) + 1 = l(v), u = s 1... ŝ i... s m ( i) for reduced expression v = s 1... s m. We induce, L on S k+1 ( C k+1 P k ). A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

14 Background Objects Example. P 3 C 4 S 4. (c() and w are displayed) s s23210 s s12310 s s 3210 s 0310 s2310 s s 210 s310 s Remark. (Facts) w w µ c() c(µ). µ = c() c(µ). w L w µ = µ. s s 0 0 e s30 A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

15 Weak strips and (K-)k-Schur functions 1 Background Objects 2 Weak strips and (K-)k-Schur functions 3 Results A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

16 Weak strips and (K-)k-Schur functions Definition (Cyclically decreasing sequence) A sequence (i 1,..., i m ) in Z k+1 = {0, 1,..., k} is cyclically decreasing if a < b s.t. i b i a or i a + 1 mod k + 1. i.e. there is no appearance of..., j,..., j,... nor in (i 1,..., i m )...., j,..., j + 1,... Definition (Cyclically decreasing element) For A = {i 1,..., i m } Z k+1 where (i 1,..., i m ) is cyclically decreasing, d A := s i1... s im ( S k+1 ). Example. k = 5. A = {0, 1, 3, 5} Z 6. d A = s 1 s 0 s 5 s 3 = s 1 s 0 s 3 s 5 = s 1 s 3 s 0 s 5 = s 3 s 1 s 0 s 5. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

17 Weak strips and (K-)k-Schur functions Weak Strips and k-schur functions Definition (weak strip) For, µ P k, µ/: a weak strip of size r : A Z k+1 s.t. A = r, w µ = d A w L w ( A Z k+1 s.t. A = r, w µ = d A w, l(w µ ) = l(d A ) + l(w ) ) Remark. In fact, µ/: a weak strip { c(µ)/c(): horizontal strip w µ L w { µ/: horizontal strip p(c(µ) )/p(c() ): vertical strip. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

18 Weak strips and (K-)k-Schur functions Weak Strips and k-schur functions Definition (k-schur function) k-schur functions s (k) h r s (k) = is characterized by s (k) = 1 and s µ (k). ( for P k, r k) µ/: weak strip of size r Remark. {s (k) } P k is Z-bases of Λ (k) = Z[h 1,..., h k ]. s (k) : homogeneous A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

19 Weak strips and (K-)k-Schur functions Example. k = 3. = P 3, c() = C 4, w = s S 4. s d {1,2} w d {1,2,3} w s 2 s d {1} w d {1,3} w s 1 s w s d {3} w e.g. {weak strips/w of size 2} = {d 1,2 w, d 1,3 w }. h 2 s (3) = s(3) d {1,3} + s(3) d {1,2}. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

20 Weak strips and (K-)k-Schur functions Weak set-valued strips and K-k-Schur functions { x (if x > s i x) For i Z k+1, let φ i : S k+1 S k+1 ; φ i (x) = s i x (if x < s i x). For x = s i1... s im (red. exp.), let φ x = φ i1... φ im. Definition (weak set-valued strip) (µ/, A): a weak set-valued strip of size r : A Z k+1 s.t. A = r, w µ = φ da (w ). Definition (K-k-Schur function) K-k-Schur functions g (k) h r g (k) = is characterized by g (k) = 1 and ( 1) A ( µ ) g µ (k). (µ/, A): weak set-valued strip of size r A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

21 Weak strips and (K-)k-Schur functions Example. k = 3. = P 3, c() = C 4, w = s S 4. s d {1,2} w d {1,2,3} w s 2 s d {1} w d {1,3} w s 1 s s1 s 2 w s d {3} w weak s-v. strips/w of size 2 (w µ, A) ( 1) A ( µ ) (w, {0, 2}) +1 (d 1 w, {0, 1}) 1 (d 3 w, {2, 3}) 1 (d 3 w, {3, 0}) 1 (d 1,2 w, {1, 2}) +1 (d 1,3 w, {1, 3}) +1 s 0 h 2 g (3) = g (3) d {1,3} + g (3) d {1,2} g (3) (3) d {1} 2g d {3} + g (3). A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

22 Weak strips and (K-)k-Schur functions Remark. {g (k) } P k in Z-bases of Λ (k) = Z[h 1,..., h k ]. g (k) : inhomogeneous, highest degree term = s(k) A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

23 Results 1 Background Objects 2 Weak strips and (K-)k-Schur functions 3 Results A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

24 Results Auxiliary Results (A): On the lattice property on W For (P, ): poset, a, b P, their meet a b := max{c P a c b} (if max exists), their join a b := min{c P a c b} (if min exists). Let W : Coxeter group, x, y W. Fact x L y = max L {z x L z L y} exists. x L y = min L {z x L z L y} exists if {z x L z L y}. Remark. x y = max {z x z y} and x y = min {z x z y} do not always exist. e.g. S 3 s 121 s 21 s 12 s 1 s 2 e A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

25 Results Auxiliary Results (A): On the lattice property on W W : Coxeter group x, y W. Lemma (T.) x S L y := max {z W x z L y} exists. x S L y := min {z W x z L y} exists. Sketch: present x = s i1... s in s ja1... s ja t (1 a 1 < < a l m) y = so that t is maximal. Then s j s jm x L S y = s ja1... s ja t x S L y = s i1... s in s j s jm A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

26 Results Auxiliary Results (B): On the weak strips Lemma (T.) { d A B /, d A B /: w.s. d A /, d B /: w.s. = d A B = (d A ) (d B ) under. where w.s. = weak strip, : strong order s 3s 2s 1 3 s 2s 1 s 3s s s 3 {1, 2, 3} 3 {1, 2} {1, 3} {1} {3} 1 3 k = 3. Weak strips over = P 3, c() = C 4, w = s S 4. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

27 Results Main Results (A) a Pieri-type formula For P k and 1 r k, let g (k) = µ g (k) µ, hr = h 0 + h h r. Let {d A1, d A2,... } be the list of weak strips over of size r. Theorem (T.) g (k) h r = = a µ d Ai ( i) g (k) µ g (k) d Aa g (k) d Aa Ab + g (k) d Aa Ab Ac.... a<b a<b<c (The second equiality is from d B C = (d B ) (d C ) and the Inclusion-Exclusion Principle) A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

28 Results Example. k = 3. Table of g (k) h i = (k) µ (coeff)g µ h 0 h 1 g (3) h1 = g (3) µ = g (3) + g (3) + g (3) 2 g (3). 0 µ or µ or µ A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

29 Results Main Results (B): k-rectangle factorization (1) k-rectangle: R t = (t,..., t) P }{{} k for 1 t k k+1 t Theorem (Lapointe, Morse) s (k) R = t s(k) R t s (k) for P k. Here, µ ν = the partition obtained by reordering (µ 1,..., µ l(µ), ν 1,..., ν l(ν) ) = R t R t A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

30 Results Main Results (B): k-rectangle factorization (2) Similar formula for g (k) {? g (k) R t g (k) R (in Z[h t 1,..., h k ]) is proved but g (k) R g (k) t g (k) in general Example. g (k) R t (r) = R t { g (k) R t g (k) R t g (k) ((r) g (3) = g (3) ( g (3) + g (3) ) g (3) = g (3) ( g (k) (r) + g (k) (r 1) + + g (k) ) (if t < r), (if t r) g (3) + g (3) + g (3) + g (3) + g (3) + g (3) + g (3) + g (3) ) A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

31 Results Main Results (B): k-rectangle factorization (3) Theorem (T.) For any P k and 1 t k, g (k) R t (k) = g R t g (k). Remark. s (k) g (k) is not a ring homomorphism. A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

32 Results Outline of the proof: Pieri Write simply W = S k+1 and W = S k+1. Recall the Pieri rule g v (k) h i = ( 1) i (l(φ d (v)) l(v)) A g (k) φ da (v). A I, A =i φ da (v) W Summing over v {v W v w} and i = 0, 1,..., r, g w (k) h r = ( 1) A (l(φ d (v)) l(v)) A g (k) φ da (w), v w A I, A r v W φ da (v) W and its coefficient of g u (k) (u W ) is ( 1) A (l(u) l(v)) = v w v W A I, A r u=φ da (v) A r v w v W u=φ da (v) ( 1) A (l(u) l(v)). A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

33 Results Outline of the proof: Pieri (cont.)... and its coefficient of g u (k) W is ( 1) A (l(u) l(v)) = v w v W A I, A r u=φ da (v) A r v w v W u=φ da (v) ( 1) A (l(u) l(v)) A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

34 Results Outline of the proof: Pieri (cont.)... and its coefficient of g u (k) W is ( 1) A (l(u) l(v)) = where v w v W A I, A r u=φ da (v) A r = A r v w v W u=φ da (v) ( 1) A (l(u) l(v)) v Y A,u ( 1) A (l(u) l(v)), Y A,u := {v W φ da (v) = u and v w}. Step 1. Y A,u is isomorphic to boolean lattices. Hence { ( 1) A (l(u) l(v)) 1 if Y A,u = 1, = 0 otherwise. v Y A,u A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

35 Results Outline of the proof: Pieri Step 2. Y A,u = 1 d 1 A u = max{z w z L u}(= w S L u). Hence { ( 1) A (l(u) l(v)) A r, = 1 A s.t. d 1 v Y A,u A u = w S L u. A r Step 3. A s.t. { A r, d 1 A u = w S L u u d C w for C s.t. { C r d C w/w: weak strip A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

36 Results Outline of the proof: k-rectangle factorization (Basically similar to the proof of s (k) R = t s(k) R t s (k) ) Define a linear map Θ : Λ (k) Λ (k) by g (k) g (k) R ( P t k). It suffices to show Θ( h r g (k) ) = h r Θ( g (k) ). (1) (since it implies Θ is Λ (k) -hom and ) g (k) R t (k) (k) (k) (k) = Θ( g ) = g Θ(1) = g Θ( g ) = g (k) (k) g R t A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

37 Results Outline of the proof: k-rectangle factorization Let {d A1, d A2,... } be the list of weak strips over of size r. From the Pieri rule for g (k), h r g (k) = a g (k) d Aa g (k) d Aa Ab +..., a<b Applying Θ to this, (LHS) of (1) is Θ( h r g (k) ) = a g (k) R t d Aa g (k) R t d Aa Ab a<b A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37

38 Results Outline of the proof: k-rectangle factorization {d A1, d A2,... } : the list of weak strips over of size r. Fact (a) {d A1 +t(r t ), d A2 +t(r t ),... } is the list of weak strips over R t of size r. (b) d Ai +t(r t ) = R t (d A ). From the Pieri rule for g (k) and Fact(a), (RHS) of (1) is h r Θ( g (k) ) = h r g (k) R t = a g (k) d Aa+t (R t ) g (k) d (Aa+t) (Ab +t)(r t ) a<b From Fact (b), this equals to (LHS) of (1). A Pieri-type formula and a factorization formula for K-k-Schur February functions 22, / 37