Alcove random walks and k-Schur functions Cédric Lecouvey and Pierre Tarrago IDP (Tours) and LPSM (Paris) FPSAC 2019 Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 1 / 21
0. General considerations There exist natural generalizations of the Young lattice. 1 Their extremal harmonic functions make appear interesting families of 2 polynomials. These harmonic functions also control the behavior of certain random walks. 3 Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 2 / 21
I. Combinatorics A partition of rank l is a nonincreasing sequence λ = ( λ 1 � � � � � λ m ) 2 Z � 0 s.t. λ 1 + � � � + λ m = l . The partition λ is encoded by its Young diagram. Each cell c of λ has a hook length h ( c ) . Let hook ( λ ) = λ 1 + d � 1 where d = max ( i j λ i > 0 g . Example | | | | | λ = ( 5 , 5 , 3 , 3 , 1 ) � with h ( 1 , 2 ) = 7, hook ( λ ) = 9 and | | transposed partition tr ( λ ) = ( 5 , 4 , 4 , 2 , 2 ) � Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 3 / 21
Fix k � 1 an integer A ( k + 1 ) -core is a partition λ with no hook length equal to k + 1 . Write j λ j k for the number of cells with hook length less or equal to k . Example The partition • • • • • • λ = • • • • is a 4-core with j κ j 3 = 10 (but not a 3-core). Observe λ is a ( k + 1 ) -core i.f.f tr ( λ ) is. Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 4 / 21
A partition is k -bounded when its parts are less or equal to k . For k …xed, there is a bijection c � f λ j k + 1-core with j λ j k = l g c � 1 f µ j k -bounded of rank l g obtained by deleting the cells with hook lengths greater than k + 1 and next left align. Example • • • • • • � λ = • • • • Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 5 / 21
The map ι = c � 1 � tr � c is an involution on the k -bounded partitions. Let Y k be the graph with vertices the k -bounded partitions and arrows λ ! µ when µ is obtained by adding one cell to λ Observe that lim k ! + ∞ Y k = Y is the Young lattice of ordinary partitions. Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 6 / 21
The map ι = c � 1 � tr � c is an involution on the k -bounded partitions. Let Y k be the graph with vertices the k -bounded partitions and arrows λ ! µ when µ is obtained by adding one cell to λ ι ( µ ) is obtained by adding one cell to ι ( λ ) . Observe that lim k ! + ∞ Y k = Y is the Young lattice of ordinary partitions. Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 6 / 21
The graph Y 2 Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 7 / 21
II. Harmonic functions A function f : Y k ! R � 0 is harmonic when f ( ∅ ) = 1 and for any λ 2 Y k f ( λ ) = ∑ f ( µ ) . λ ! µ Positive harmonic functions parametrize central Markov chains on Y k : the transition matrix associated to f is Π ( λ , µ ) = f ( µ ) f ( λ ) 1 λ ! µ and Π ( λ ( 1 ) , λ ( 2 ) , . . . , λ ( l ) ) = f ( λ ( l ) ) f ( λ ( 1 ) ) only depends on the ends of the trajectory λ ( 1 ) , λ ( 2 ) , . . . , λ ( l ) . Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 8 / 21
Problem (Minimal boundary of Y k ) What are the extremal nonnegative harmonic functions on Y k ? The graph Y k is multiplicative : there exists a R -algebra A with a distinguished basis B = f s ( k ) j λ 2 Y k g s.t. λ s ( k ) = 1 ∅ Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 9 / 21
Problem (Minimal boundary of Y k ) What are the extremal nonnegative harmonic functions on Y k ? The graph Y k is multiplicative : there exists a R -algebra A with a distinguished basis B = f s ( k ) j λ 2 Y k g s.t. λ s ( k ) = 1 ∅ s ( k ) λ s ( k ) = ∑ λ ! µ s ( k ) µ 1 Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 9 / 21
Problem (Minimal boundary of Y k ) What are the extremal nonnegative harmonic functions on Y k ? The graph Y k is multiplicative : there exists a R -algebra A with a distinguished basis B = f s ( k ) j λ 2 Y k g s.t. λ s ( k ) = 1 ∅ s ( k ) λ s ( k ) = ∑ λ ! µ s ( k ) µ 1 s ( k ) λ s ( k ) decomposes on B with nonnegative coe¢cients (only a geometric µ proof by Lam 2008). Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 9 / 21
Theorem (Kerov-Vershik 1989) The extremal harmonic functions on Y k correspond to the morphisms θ : A ! R s.t. θ ( s ( k ) ) = 1 and θ ( s ( k ) λ ) � 0 for any λ 2 Y k by setting 1 f ( λ ) = θ ( s ( k ) λ ) Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 10 / 21
III. k-Schur functions Let Λ = Sym R ( x 1 , . . . , x n , . . . ) the algebra of symmetric functions. Recall that the ∑ h a = x i 1 � � � x i a 1 � i 1 ����� i a with a � 1 algebraically generate Λ Set A = h h 1 , . . . , h k i and the s ( k ) λ , λ 2 Y ( k ) are the k -Schur functions of Lapointe, Lascoux and Morse (2003). They are de…ned from the multiplication s ( k ) � h a ( k -Pieri rule) which is encoded λ in Y ( k ) We have lim k ! + ∞ s ( k ) = s λ the Schur function associated to λ . λ Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 11 / 21
When hook ( λ ) � k , we have s ( k ) = s λ . λ λ 2 Y k is k -irreducible when λ does not contain any rectangle R a = ( k � a + 1 ) � a for a = 1 , . . . , k . Theorem (Lapointe, Morse (2007)) For any λ 2 P ( k ) , there is a unique factorization s ( k ) = s p 1 R 1 � � � s p k R k s ( k ) κ λ with κ 2 P ( k ) irr (the set of k-irreducible partitions). Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 12 / 21
Example For k = 3 we have R 1 = , R 2 = , R 3 = and for z z z ~ ~ ~ ~ λ = ( 3 , 2 , 2 , 2 , 1 , 1 ) = � � � � s ( 3 ) = s ( 3 ) s ( 2 , 2 ) s ( 3 ) ( 2 , 1 , 1 ) . λ Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 13 / 21
We have card ( P ( k ) irr ) = k ! Example For k = 3, there are 6 irreducible 3-restricted partitions ∅ , , , , , . Thus, the relevant morphisms θ are those s.t. 8 θ ( s 1 ) = 1 < θ ( s R a ) � 0 for any a = 1 , . . . , k : θ ( s ( k ) ) � 0 for any κ 2 P ( k ) irr . κ Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 14 / 21
From the rectangle factorization, one can write for any κ 2 P ( k ) irr s ( k ) � s ( 1 ) = ∑ s ( k ) m κ , κ 0 ( s R 1 , . . . , s R k ) s ( k ) ∑ = κ µ κ 0 κ ! µ κ 0 2P ( k ) irr where m κ , κ 0 ( s R 1 , . . . , s R k ) 2 Z � 0 [ s R 1 , . . . , s R k ] de…ne a k ! � k ! matrix M k ( s R 1 , . . . , s R k ) . Theorem (L, Tarrago (2018)) By specializing s R a = r a 2 R � 0 , M k ( r 1 , . . . , r k ) is irreducible i.f.f r a + r a + 1 > 0 for any a = 1 , . . . , k � 1 . Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 15 / 21
Example For k = 3, on gets 0 1 0 0 s R 1 s R 3 s R 2 0 B C 1 0 0 0 0 s R 2 B C B C 0 1 0 0 0 s R 3 B C M 3 = B C 0 1 0 0 0 s R 1 B C @ A 0 0 1 1 0 0 0 0 0 0 1 0 Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 16 / 21
IV. Primitive element theorem Remind R = R [ s R 1 , . . . , s R k ] is a subalgebra of A . Let L and K be the fraction …elds of A and R , respectively. Theorem (L, Tarrago (2018)) L is a separable and algebraic extension of K of degree k ! . Moreover L = K [ s 1 ] i.e. s 1 is a primitive element in L . Corollary There exists a polynomial ∆ 2 R s.t. each polynomial s ( k ) with κ 2 P ( k ) irr can be κ written on the form = 1 s ( k ) ∆ P κ ( s 1 ) κ with P κ 2 R [ X ] . Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 17 / 21
V. Reduced alcove walks of type A For i = 1 , . . . , k � 1 set α i = e i � e i + 1 in R k and put α 0 = � ( α 1 + � � � + α k ) . There is a tesselation of R k by alcoves supported by the hyperplanes H i , m = f v 2 R k j ( v , α i ) = m g with i = 0 , . . . , k � 1 and m 2 Z . The dominant alcoves are those in the cone delimited by the hyperplanes H α _ i = H i , 0 . Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 18 / 21
Figure: A reduced walk on dominant alcoves for k = 2 Lecouvey-Tarrago (IDP (Tours) and LPSM (Paris)) Alcove random walks and k-Schur functions FPSAC 2019 19 / 21
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